  We prove that the spectral radius of a large random matrix $X$ with
independent, identically distributed complex entries follows the Gumbel law
irrespective of the distribution of the matrix elements. This solves a
long-standing conjecture of Bordenave and Chafa{\"\i} and it establishes the
first universality result for one of the most prominent extremal spectral
statistics in random matrix theory. Furthermore, we also prove that the
argument of the largest eigenvalue is uniform on the unit circle and that the
extremal eigenvalues of $X$ form a Poisson point process.
