Intended mainly for use as a reference manual, this third edition encompasses all the improvements of the newest version of the PLTMG software package. This updated version introduces several significant changes, including the following: The data structures defining the domain (triangulation and skeleton) have been simplified; more general boundary conditions are allowed; the internal data structure (triangle tree) has been replaced by a simple triangulation; the adaptive algorithms have been revised to use this simple data structure; the linear equations solvers are multigraph methods, which are simple ILU factorizations with strong connections to the hierarchical basis multigrid method; the graphics packages have been improved and now include expanded options and the test driver has a new X-Windows gui option.