Din Tai Fung, Taipei
This place was apparently named by the NY Times as being one of the Top 10 restaurants in the world. The hsiao long bao (literally, little basket dumplings) is the thing that made them famous, particularly due to the painstaking labor that you can watch them go through in the glass window when doing on the little folds in each dumpling by hand. The skin is very thin too, which is the mark of distinction for quality - just be careful when picking it up so as not to poke through and let all the broth out (try to master the spoon and hole-bite technique).
I used to like this place a lot a long time ago. But upon my first visit to Shanghai, I realized that the Shanghainese still do it better as the meat inside is tastier (I suspect more pork lard??). The place I went to in Shanghai definitely didn't show the delicate craftsmanship of Din Tai Fung as the skin was a bit thicker and didn't have all those little folds, but the taste was a little better.
Still, Din Tai Fung definitely is far from bad. In fact, I do like the many other dumplings they have, including some of the chive and mushroom stuffed ones, as well as their great chicken soup. Din Tai Fung has spread quite rapidly to other locations, including Japan, LA, and Singapore. Some would argue that the other branches just don't taste as good as the original ones in Taipei. I can taste a little bit of a difference in the skin and ingredients, but I'd still just as easily go to the other branches. Just watch out for the long lines (I was surprised to see the other night that the original location on Hsin Yi Road was still packed - I thought the crowds has dissipated to the other Taipei locations long ago).