We present a fast vortex method for incompressible flow in three dimensions. It is based on two ideas. The first is an extension of Anderson's Method of Local Corrections (MLC) algorithm, a particle-particle particle-mesh method, from two spatial dimensions to three. In this approach, the calculation of the velocity field induced by a collection of vortices is split into two parts: (i) a finite difference velocity field calculation using a fast Poisson solver, the results of which are used to represent the velocity field induced by vortices far from the evaluation point; and (ii) an N-body calculation to compute the velocity field at a vortex induced by nearby vortices. The second idea is the use of adaptive mesh refinement in the finite difference calculation. Calculations with a vortex ring in three dimensions show the break-even point between the MLC with AMR and the direct method is at N approximately 3000 on a Cray Y-MP; for N approximately 64,000, MLC with AMR can be twelve times faster than the direct method. Results from calculations of two colliding inviscid vortex rings are presented; these results demonstrate the increased resolution which can be obtained using fast methods.