Who is the last mathematician that understood all of mathematics. We were discussing this question at dinner this evening: Who is the last mathematician who had an understanding of a large proportion of mathematics (at the time they were alive)? 
I think it is safe to say that the mathematician lived and worked before the secong world war. After that period it has become impossible to aquire knowledge of such a large and rapidly growing subject. 
Picard (1856-1941) was the best suggestion we have come up with because he published research papers and wrote text books in a vast range of different mathematical subjects. 
 A: Poincaré is often mentioned in that light.
A: I can only attest that it is common folklore that Hilbert is the last mathematician to have understood all of mathematics (I can't recall where I've read this; but I know I've seen or heard this in more than one place). 
But this is folklore; I of course can't judge whether this is true, or if other mathematicians (you mention Picard, and Joel David Hamkins even says von Neumann) truly fit this criteria. I feel this is very subjective and the only people who may be allowed to make such claims should be experts very knowledgeable in the history of mathematics.
EDITED:
Picard was born 6 years before Hilbert, and they both died in the early 40's. So chronologically it would make sense if they both represent the last generation that could have had a full understanding of all of mathematics.
A: Many people have mentioned Jean Dieudonné (1906-92) in this regard.
A: By the way, facetiously one could add "Bourbaki" to that list.
A: John von Neumann. 
From Wikipedia:


John von Neumann (December 28, 1903 – February 8, 1957) was a Hungarian-American mathematician who made major contributions to a vast range of fields,[1] including set theory, functional analysis, quantum mechanics, ergodic theory, continuous geometry, economics and game theory, computer science, numerical analysis, hydrodynamics (of explosions), and statistics, as well as many other mathematical fields. He is generally regarded as one of the greatest mathematicians in modern history.[2] The mathematician Jean Dieudonné called von Neumann "the last of the great mathematicians",[3] while Peter Lax described him as possessing the most "fearsome technical prowess" and "scintillating intellect" of the century.[4] Even in Budapest, in the time that produced geniuses like von Kármán (b. 1881), Szilárd (b. 1898), Wigner (b. 1902), and Edward Teller (b. 1908), his brilliance stood out.[5]
Von Neumann was a pioneer of the application of operator theory to quantum mechanics, in the development of functional analysis, a principal member of the Manhattan Project and the Institute for Advanced Study in Princeton (as one of the few originally appointed), and a key figure in the development of game theory[1][6] and the concepts of cellular automata[1] and the universal constructor. Along with Teller and Stanislaw Ulam, von Neumann worked out key steps in the nuclear physics involved in thermonuclear reactions and the hydrogen bomb.


My own personal favorite of his ideas are the von Neumann ordinals: every ordinal number is precisely the set of smaller ordinals.
