In this work, we develop Lp boundedness theory for pseudodifferential operators with rough (not even continuous in general) symbols in the x variable. Moreover, the B(Lp) operator norms are estimated explicitly in terms of scale invariant quantities involving the symbols. All the estimates are shown to be sharp with respect to the required smoothness in the ξ variable. As a corollary, we obtain Lp bounds for (smoothed out versions of) the maximal directional Hilbert transform and the Carleson operator. © Birkhäuser Boston 2009.