This article is concerned with local well-posedness of the Cauchy problem for second order quasilinear hyperbolic equations with rough initial data. The new results obtained here are sharp in low dimension. 1. Introduction . The results. We consider in this paper second order, nonlinear hyperbolic equations of the form () gij (u) ∂i ∂j u = q ij (u) ∂i u ∂j u on R × Rn , with Cauchy data prescribed at time 0, () u(0, x) = u0 (x) , ∂0 u(0, x) = u1 (x) .