Vortex methods are methods for solving time-dependent incompressible flow problems by discretizing the vorticity into vortex elements and following these elements in time. The main difficulty with vortex methods as originally formulated is that the cost of the evaluation of the velocity field induced by N vortices is O(N^2). This is expensive, particularly in three dimensions where the number of elements can increase rapidly in time due to vortex stretching. In this paper, 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 in two dimensions, to the case of three dimensional problems. In this approach, the calculation of the velocity field induced by a collection of vortices is split into two parts: (i) 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 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 breakeven point between the MLC with AMR and the direct method is at N approximately 3000; for N approximately 64,000, MLC with AMR can be twelve times faster than the direct method.