  We obtain Calder\'on-Zygmund type estimates in generalized Morrey spaces for
nonlinear equations of $p$-Laplacian type. Our result is obtained under minimal
regularity assumptions both on the operator and on the domain. This result
allows us to study asymptotically regular operators. As a byproduct, we obtain
also generalized H\"older regularity of the solutions under some minimal
restrictions of the weight functions.
