Although the history of fractional order derivatives is nearly as long as that of regular (integer order) derivatives, their range of applications has increased dramatically in recent decades. In contrast to an integer order derivative, a fractional order derivative is not a local operator. It is most often expressed as an integral between some “base point” and an “evaluation point.” Although this integral is singular at least at the evaluation point, the FD perspectives provided in the previous chapters can still be applied and have recently led to a very high-order accurate method for their approximation. This is the case both for purely real-valued functions and for analytic functions in the complex plane. Fractional order Laplace operators are also frequently encountered in various applications, and methods for their approximation are discussed.