The authors consider the Hodge Laplacian on the Heisenberg group Hn, endowed with a left-invariant and U(n)-invariant Riemannian metric. In this paper they address three main, related questions: (1) whether the L2 and Lp - Hodge decompositions, 1 < p < 8, hold on Hn, (2) whether the Riesz transforms dk¿ are Lp-bounded, for 1<<8 (3) how to prove a sharp Mihilin-Hormander multiplier theorem for k, 0 = k = 2n 1.