In the last 20 years, a vast variety of algorithms have been developed which are based on the concept of complementary pivoting. Many of these are listed in our bibliography. The PL continuation and homotopy algorithms described in the last two chapters are important examples. In order to give a better idea of the flexibility which is possible and to describe the construction of such algorithms for special purposes, we are now going to cast the notion of PL algorithms into the more general setting of PL manifolds. Eaves (1976) has given a very elegant geometric approach to general PL methods which has strongly influenced the writing of this chapter, see also Eaves & Scarf (1976). In the first two sections we give a general formulation of PL algorithms in the context of PL manifolds which will then allow us to describe and study a variety of sophisticated PL algorithms in a unified framework.