  We prove regularity estimates for weak solutions to linear nonlocal
equations. The nonlocal operators under consideration are integro-differential
operators with almost no order of differentiability. Under rather weak
assumptions, which allow the operator to be anisotropic and weakly singular, we
show that locally bounded weak solutions are continuous. We prove uniform
a-priori estimates for the modulus of continuity. Furthermore, we provide
sufficient conditions for the boundedness of weak solutions.
