We consider the Aluthge transform of a Hilbert space operator , where is the polar decomposition of . We prove that the map is continuous with respect to the norm topology and with respect to the -SOT topology on bounded sets. We consider the special case in a tracial von Neumann algebra when implements an automorphism of the von Neumann algebra generated by the positive part of , and we prove that the iterated Aluthge transform converges to a normal operator whose Brown measure agrees with that of (and we compute this Brown measure). This proof relies on a theorem that is an analogue of von Neumann's mean ergodic theorem, but for sums weighted by binomial coefficients.