  In this paper we establish for the first time local semiconvexity estimates
for fully nonlinear equations and for obstacle problems driven by
integro-differential operators with general kernels. Our proof is based on the
Bernstein technique, which we develop for a natural class of nonlocal operators
and consider to be of independent interest. In particular, we solve an open
problem from Cabr\'e-Dipierro-Valdinoci [CDV22]. As an application of our
result, we establish optimal regularity estimates and smoothness of the free
boundary near regular points for the nonlocal obstacle problem on domains.
Finally, we also extend the Bernstein technique to parabolic equations and
nonsymmetric operators.
