We present recent results on nonlocal operators acting on real-valued functions that are defined on subsets of . The operators under consideration exhibit a fractional order of differentiability and have attracted a lot of attention in the last twenty years. It turns out that several ideas and approaches developed for the study of partial differential operators of second order can be applied after suitable modifications. In this report, we explain similarities and differences with regard to the well-established theory for differential operators of second order. We concentrate on stationary linear symmetric operators in variational form.