I don't actually swear off plants all that often. True, there have been a few that pushed me too far, but generally I'm optimistic enough that I stop short of ruling a plant out forever and ever. Originally, when I started writing this, Streptocarpus was going on the list too.

Compared to others on the list, it's not that bad, really. It's not especially buggy, like Hedera helix or Codiaeum variegatum, and it's not perpetually miserable like Philodendron 'Xanadu,' or a dramatic and heartbreaking failure like Calathea ornata. I've just never had a good experience with one, either at home or at work. Or, well, I had one good experience, with one individual plant. It lasted about two months. Then it turned into another bad experience.
My first experience with Streptocarpus was in the fall of 2007, right after starting the greenhouse job. Apparently people here treat streps like annuals and keep them outside, and then pitch them in the fall. Or that was the theory: I've been watching, but have yet to see anybody with one on their front porch or planted in a flower bed. Who knows what they're doing with them really. Nevertheless: if they come in, they come in in the spring.1 And in August 2007 when I started, there were still a lot of those left over from the spring, on the east side of the building, getting cooked in the morning August sun. Which was kind of dumb, because they don't like direct August sun. (I never claimed we were smart.2)
So we brought them into the greenhouse (looking back at it now, this was probably a bad call: however hot the east side of the building got, the greenhouse got hotter), and then I noticed that some of them had mealybugs. So we pulled them out of the greenhouse into the back room, and squished the visible bugs, and sprayed all kinds of crap on the leaves to kill the invisible bugs. Which left big ugly black spots on the leaves. Not helpful. And the mealybugs persisted anyway, so we were scarring the leaves for nothing. So they got dumped. I don't know how many we threw out. It seemed like a lot at the time.
And then I didn't see any Streptocarpus again until the spring of 2008. We got a bunch of plugs in, and potted them up, and although a few rotted right away, that was fine: we had plenty. And when they started blooming, I thought they were pretty. So I picked out a nice one, all loaded up with buds, of a red-pink variety called 'Tanager,' and took it home. Whereupon it immediately dropped all of its buds and then sat there, sulking, for about six months, doing absolutely nothing.

The plants that stayed behind at work weren't doing any better: the greenhouse got hot, so we moved them outside, and then it rained, and rained, and rained, and flooded, so all of them rotted and died as a result.3
Meanwhile, around September, my 'Tanager' at home suddenly started growing leaves again. Which was great. I mean, leaves aren't as good as flowers, but whatever. I was just thrilled it was doing something. And then come November, we had flowers too. It was great.
It was also temporary: that plant was dead by January 2009. It just kind of went limp one day and wouldn't come back.
So I tried again, in March, this time with the cultivar 'Purple Martin.'4 It didn't come to a complete stop once I got it home, like 'Tanager' had: it continued to grow leaves, and it even flowered once or twice.
And then it all went limp at once and wouldn't revive when watered and it was dead by June. I lost a lot of plants between May and June, because we moved to a new house and everything was chaotic, including waterings, but still.5

So I've seen Streptocarpus die in all kinds of interesting and unexpected ways, and I'm not a fan, and I don't recommend giving them to anybody unless you're trying to break down their self-esteem or something.6 I'm not willing to rule them out entirely, because in the process of writing this profile I've seen some pictures of some really ohmygod beautiful ones, and for the right plant, I might be willing to try again. But we'd have to be talkin' about one charming motherfucking pig Streptocarpus.
But enough about me. You're probably here because you want to know how to care for them, right? Crap. Um. . . .
The other websites make them sound easy enough. Basically like African violets, but easier, is the impression I get from reading around. Here's the general consensus from the net, with particular attention given to places like streptocarpus-info.com and robsviolets.com,7 which are trying very hard to give the impression that they know what they're doing, and maybe actually do:
LIGHT: Bright light, with some sun, appears to be ideal, as for example an east or west window. Too much sun will burn the leaves, and they won't flower if they're not getting enough light. One way around both problems is to grow them under fluorescent lights: like African violets, Streptocarpus do well in artificial light.

WATERING: Supposedly, Streptocarpus do best if allowed to dry out between thorough waterings, even to the point of going a little limp (like for Spathiphyllum), though they should never be allowed to stand in water or dry out completely. This is more or less the way I water all of my plants, with a few exceptions, so you'd think I would have been able to manage this, but no. It's also not as simple as that, because as with most other gesneriad species, if you get cold water on the leaves, the leaves will develop spots. (You can bottom-water to get around that, or just use room-temperature water.)
The bigger issue, though, seems to be with soil quality: Streptocarpus must have an extremely light and porous soil or they will develop root rot. (This is my best guess for what went wrong with my plants.) The exact composition varies depending on who is doing the recommending, but both streptocarpus-info.com and robsviolets.com advise a minimum of 50% perlite.

Plants are best in wide, shallow pots than in tall, skinny ones. Azalea pots (which are 3/4 as tall as they are wide) are usually what is recommended, though this is more of a gentle recommendation than a hard and fast rule. The boss's mother at the greenhouse had her plants in pots that were deeper than they were wide, and they were the most amazing Streptocarpus I'd ever seen.
A number of sites repeat the same advice over and over, to water the plant when the top 1/2 inch of soil is dry. My advice would be, don't rely too much on that as a guideline, because the moisture level in that top half inch will indicate very different things in dense, wet soil than it will in light, airy soil. Waiting to see the leaves go slightly limp seems more reliable to me.8
HUMIDITY: Humidity levels have a large influence on certain other aspects of care, more so than with a lot of other genera of houseplant. If the plant is in high humidity, it can take more direct sun, and for a longer period, than a plant which is in low-humidity air. They are also more tolerant of underwatering and high temperatures if the air is moist. As a general rule, more humidity is better, the one exception being that tightly crowded plants in high humidity are prone to rot and fungus. So don't increase humidity at the expense of air circulation.
TEMPERATURE: Everybody is in agreement that Streptocarpus do best in a cool room. Definitely keep the temperature below 80F (27C) as much as possible, and aim for somewhere in the range between 55-75F (13-24C). I can more or less vouch for this personally: in the greenhouse, our plants did well if they were near the doors (where they occasionally got cool air) and began to droop and look miserable at about the same temperatures I did.9

PESTS: Mealybugs are the only ones I've seen personally, and appear to be the most common problem, though thrips and mites, I'm told, can also affect Streptocarpus. None of the sites I ran into seemed to really emphasize pests as an issue, though, so I expect that cultural issues are the main cause of death. (I should be so lucky, to keep a strep alive long enough for it to get bugs.)
PROPAGATION: I've never managed to make this work out, either, but supposedly these can be propagated much like Saintpaulia or Begonia. The usual recommendation is to take a long, healthy, middle-aged leaf, and make two parallel lengthwise cuts, one just above the midrib and one just below, and then plant the leaves in moist potting soil or vermiculite, with the cut edges in the soil and the original leaf margins sticking up. In theory, this is supposed to result in new plantlets forming at every vein along the cut leaf.
One is also supposed to be able to cut leaves into chevron-shaped sections and plant those in potting soil or vermiculite, and get new plants that way. I've tried both of these methods and neither of them resulted in new plants, though in fairness propagation wasn't really a priority and I wasn't paying terribly careful attention to the conditions. Wedge cuttings are supposed to be more reliable, but the cut-out-the-midrib method is (theoretically) capable of producing more plants, total.

Sticking leaf sections into shallow water, provided that one has a way of keeping them upright, might also be effective, according to this thread at Garden Web. Cutting out the midribs, wrapping the resulting half-leaves in damp paper towels, and enclosing the paper towels in a Ziploc bag, instead of planting them in vermiculite, is also said to work. I've not tried either of these methods at all.
Plants may also be grown from seeds. The seed pods are long, thin, and twisted brown things which are occasionally seen on plants in retail situations. (I saw them at work a time or two; I don't know why I didn't get a picture but I'm sure I must have had a very, very good reason.) I have never seen the seeds directly, but they are supposed to be extremely small, and should sprout under more or less the same conditions as fern spores: fill one side of a clear clamshell-type container with damp vermiculite, sprinkle seeds over the top, don't cover the seeds, close the container and leave it in a bright spot until seeds begin to germinate. I don't know of any vendors who sell Streptocarpus seeds, but I'm sure there must be some out there somewhere.

One may also divide mature plants with multiple crowns; this has the advantage of providing you, in theory, with new plants that already have root systems, but the disadvantage is that it will make you feel like a big, lumbering brute who destroys anything s/he touches, as the stems, leaves, and roots are all very fragile. The plants don't necessarily care about this, though, and divisions will happily re-root in a bright, humid environment even if you didn't manage to retain any roots while dividing.
GROOMING: Grooming is actually kind of a bigger pain than I'd originally thought. Particularly in low humidity, leaves will develop brown edges and tips, which don't really hurt anything but which people generally cut off anyway. Also dead flowers need to be cleared out regularly, if you're lucky enough to get flowers, because otherwise they can begin to rot, which may lead to healthy leaves. These are not especially big issues. But:
Plants also fill their pots with roots very rapidly, in the improbable event that they're happy with you, which means that repotting and/or dividing is necessary for proper maintenance. Exactly how often this needs to happen will vary, but most of the sites I looked at that mentioned this said either every six to twelve months. That's a lot of repotting. The typical recommendation seems to be to move plants up to larger pots until they hit a 5-6" pot, and then divide plants and keep putting them back in 5-6" pots thereafter. I don't know what happens if you keep moving up to larger and larger pots. Probably they die. Dying seems to be a skill. At work, we did plant up some hanging baskets with four plugs to a ten-inch hanging basket -- some of these did fine, though what usually happened was that one or two of the plants would die, leaving the basket lopsided. A ten-inch hanging basket in which all four plants survived, particularly when all four are blooming, is definitely pretty, though, so I sort of understand going to the trouble anyway, even if it rarely worked out well enough to justify the work.

Flushing soil out with water from time to time is also sort of a big deal, apparently, though I'm a little surprised by this because it seems like you wouldn't have the plant in the same soil long enough for fertilizer salts to build up to the point of being a problem. Apparently not, though.
FEEDING: The people who are serious about growing streps recommend a high-phosphorous fertilizer from spring until fall, and then either no or very little fertilizer from fall to spring. The precise formulation doesn't seem to be particularly important: I saw recommendations for 7-9-5 and 10-30-10.10
The "person" to go along with Streptocarpus, air traffic controller, wasn't chosen because air traffic controllers sit around all day doing nothing at all, then suddenly fly into a panic and collapse, like the plant does. (At least, your better air traffic controllers don't.)
On the contrary, real air traffic controllers have to be unusually calm, and able to keep a clear head in stressful situations, and it's a difficult job to get in part because there are tests in place to weed out the people who are likely to fly into a panic and collapse. So the temperament could hardly be more wrong for this plant. However: Streptocarpus are actually one of the better houseplant examples of the phenomenon of "nectar guides," which does work as an air traffic controller reference (or at the very least it's close enough for PATSP). We'll pick up on Thursday with a discussion of what nectar guides are and why they matter. Jokes and light refreshments will be provided. Those of you who already know about nectar guides, please try not to spoil it for everybody else.
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Photo credits: all my own except for the Streptocarpus fruit picture, which came from LucaLuca via Wikipedia. I would have liked to include pictures of other varieties, but I've only ever directly encountered four ('Tanager,' 'Purple Martin,' 'Falling Stars,' 'Snow Bunting'), one of which ('Snow Bunting') photographs badly, and I couldn't find public domain / Creative Commons pictures I liked. There will be a picture of 'Falling Stars' in Part II.