The boundary element method (BEM) is applied to map the current response of the scanning electrochemical microscope for a range of tip and substrate geometries. Simulations are presented that quantify the diffusional fields around tip electrodes of disk, hemispherical, and cone geometries. Two-dimensional (axisymmetric) simulations examine the effect of the current flowing at the tip electrode as it is brought toward conducting and nonconducting surfaces that are either infinitely flat or spherically distorted. Three-dimensional BEM simulations probe the current response for approach curves where the tip microdisk electrode is not parallel to the substrate surface. The BEM was also applied to simulate a line scan using a microdisk electrode as it is positioned at various points across the surface of a substrate containing a flat macroelectrode. Finally, the three-dimensional routines were employed to produce an image of a single microdisk electrode operating in positive feedback mode embedded in a flat nonconducting substrate. Unlike previous simulations in the research area of scanning electrochemical microscopy the reduction in dimensionality derived by application of the BEM results in a considerable simplification of the grid generation procedures and a substantial reduction in simulation time required. In addition the flexibility of the BEM enables unusual substrate geometries to be addressed that would present considerable difficulties to standard finite difference procedures.