Abstract. We study the regularity of the bilinear maximal operator when applied to Sobolev functions, proving that it maps W 1,p (R) × W 1,q (R) → W 1,r (R) with 1 < p, q < ∞ and r ≥ 1, boundedly and continuously. The same result holds on R n when r > 1. We also investigate the almost everywhere and weak convergence under the action of the classical Hardy-Littlewood maximal operator, both in its global and local versions.