Researchers at U.C. Berkeley have built a robot that can fold towels. Veeeeeeerrrrrryyyyyy sssslllloooowwwwllllyyyyy. But it actually works. Now it's a mere matter of optimization.
Seriously, I would have given you even odds that this problem would not have been solved at all in my lifetime. It's one bet I would have been happy to lose, and I'm even happier that I didn't actually make it. But folding laundry -- indeed manipulating any kind of non-rigid object -- is really, really hard.
Next challenge: a robot that can tie shoelaces. To get some appreciation for how hard this is, try doing it with gloves on to simulate being a robot that doesn't have touch-sensitive skin, which most robots, including the Berkeley laundry folder, don't have. I'll bet a case of wine that this will still be an unsolved problem in 2020. Any takers?