  In this paper we establish optimal regularity estimates and smoothness of
free boundaries for nonlocal obstacle problems governed by a very general class
of integro-differential operators with possibly singular kernels. More
precisely, in contrast to all previous known results, we are able to treat
nonlocal operators whose kernels are not necessarily pointwise comparable to
the one of the fractional Laplacian. Such operators might be very anisotropic
in the sense that they "do not see" certain directions at all, or might have
substantial oscillatory behavior, causing the nonlocal Harnack inequality to
fail.
