Not a whole lot. YET!
The air conditioning switch doesn't work.
Had to replace brakes at 23000, even though most of the miles were driven on the highway.
I have to pay attention to my transmission, to adjust the pressure too much on the gas pedal for it to shift.
Tires blew at 20000 miles. Must be because its Hankok!
I will never buy a Korean car again. It has a good warranty just because they want to buy a customer. I have to play a lot with the gas pedal for the tranny to work properly, this way I don't enjoy the ride. It was supposed to be an economical car, but I make maybe 15 mpg in the city which is pretty bad.
Even though nothing big went down, I have a feeling like I'll be spending a lot of time at the dealership in the near future.
Don't dismiss Korean cars as a whole yet. Between Daewoo and Hyundai, the latter appears to be more dependable. Just like buying Korean appliances -- Samsungs I've observed to be consistently reliable, LG's (Goldstar) don't make the grade.
LG (and GoldStar) is highly reliable. We have a GoldStar Over-the-Range microwave oven, and LG 17" monitor. Both have worked flawlessly. I have also seen 2 GoldStar monitors from the late 80's being used almost daily.
I'd say if your buying a Korean car, go for a Hyundai. They're extremely well-built cars for the money. The alternative makes (Daewoo, Kia) cut corners and produce inexpensive cars with bad quality. (I'd choose Honda over any of these, but hey we can't all afford one)
As for not being able to afford a Honda, that's true if you want a new one. But you can buy a late 80s or early 90s Honda, very cheap, and it will still be more reliable than a Daewoo. I would put my money on its reliability over a new Hyundai, too. But to be fair, all indications are that Hyundais are much better than they used to be.
Any Honda from the 80's or early 90's that is "very cheap" is also going to be very high (150K+) miles. Even Hondas start to need work at that point and I doubt that one would be more reliable than a newer lower mileage Korean car. Btw, Kia is owned by Hyundai so would expect quality (or lack of) to be comparable.