  We prove fine higher regularity results of Calder\'on-Zygmund-type for
equations involving nonlocal operators modelled on the fractional $p$-Laplacian
with possibly discontinuous coefficients of VMO-type. We accomplish this by
establishing precise pointwise bounds in terms of certain fractional sharp
maximal functions. This approach is new already in the linear setting and
enables us to deduce sharp regularity results also in borderline cases.
