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have been simplified; more general boundary conditions are allowed.The internal data structure (triangle tree) has been replaced by a simple triangulation; the adaptive algorithms have been revised to use this simple data structure.The linear equations solvers are multigraph methods, which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.The test driver has a new X-Windows gui option.