Broadly applicable numerical algorithms for mapping boundaries of manipulator accessible output sets are developed and illustrated. Accessible output sets for planar and spatial manipulators are defined and analytical criteria determining their boundaries are stated, for both manipulators having the same number of input and output coordinates and redundantly controlled manipulators with a larger number of inputs than outputs. A method is presented for finding an initial point on the boundary of the accessible output set. From this point, a continuation method is used to map a family of one dimensional solution curves on the boundary. A method for finding tangents to solution curves at bifurcation points of continuation equations is presented and a computational implementation in the form of an experimental computer program is outlined. A planar redundantly controlled serial manipulator, a planar Stewart platform, and a spatial Stewart platform are analyzed, determining both the exterior boundary of the accessible output set and interior curves that represent local impediments to motion or controllability.