Consider a simple random walk on the planar lattice Z^2. We prove the Erdos-Taylor conjecture concerning the asymptotics of the number of visits to the most visited site after n steps. We also establish the asymptotics of the number of steps needed to visit every site on the nXn lattice torus. Along the way we obtain the multifractal spectrum both for frequently visited points and for those which are not visited for a long time.