This paper deals with the complexity of computing Nash and correlated equilibria for a finite game in normal form. We examine the problems of checking the existence of equilibria satisfying a certain condition, such as "Given a game G and a number r, is there a Nash (correlated) equilibrium of G in which all players obtain an expected payoff of at least r?" or "Is there a unique Nash (correlated) equilibrium in G?" etc. We show that such problems are typically "hard" (NP-hard) for Nash equilibria but "easy" (polynomial) for correlated equilibria.