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In my opinion
Windows-First, where most other free-software libraries are
Linux-First. However the quality under Linux has always been good. Since this software got reveiwed by 20 people who did not participate in its development. Speed under Linux with multiple threads has been rapidly improved around the time the asker asked this question (2009): http://think-async.com/Asio/LinuxPerformanceImprovements
Speed under Windows has always been good. My biggest gripe is the design of UDP sockets, it is poorly implemented.