Thanks for asking such an easy one :-)
Others have already answered most of it, but as it is asked from me here's my take :
As Mil said, Psion PDAs (series 3 and 5) actually were wildly successful, especially here in Europe, despite being very expensive. They still have a rabid fan base today, that will not use anything else. The Zaurus line, despite its age, is still very popular in Japan ; Sharp just couldn't be bothered to market it anywhere else. And Palm, of course, was top gun for a long time in its day.
The real sense of your question, I guess, and a very relevant one, is "where are PDAs today" ? My opinion is that PDA have faltered as a market because they were very specialized devices - they mostly did one thing right, the PIM part. Multimedia was so-so, Net access too, etc.
But that doesn't mean PIM functions have suddenly become irrelevant, just that users want devices that do many things. The dominant PIM devices today are smartphones, who also do music, video, GPS nav and of course voice.
See what I'm getting to ? ITs are all the rage now, but if they stick to their "I'm just an IT" motto, they could end up where the PDAs are now, soon enough, when more multi-purpose devices catch up...