  Pointwise estimates for the gradient of solutions to the $p$-Laplace system
with right-hand side in divergence form are established. They enable us to
develop a nonlinear counterpart of the classical Calder\'on-Zygmund theory in
terms of Calder\'on-Zygmund singular integrals, for the Laplacian. As a
consequence, a flexible, comprehensive approach to gradient bounds for the
$p$-Laplace system for a broad class of norms is derived. In particular, new
gradient estimates are exhibited, and well-known results in customary function
spaces are easily recovered.
