Many convex optimization problems can be transformed into a complementarity framework. This framework is popularly known as the complementarity problem in the literature on optimization, and it is based on the complementary slackness (CS) principle. CS principle is useful for developing pivoting algorithms, and it is based on the Karush-Kuhn-Tucker principle of optimality. For a class of convex optimization problems, Lemke proposed a pivoting algorithm called the complementary pivot algorithm for solving linear complementarity problem \((\textrm{LCP})\) . The algorithm presented by Lemke and Howson to compute an equilibrium pair of strategies to a bimatrix game, later extended by Lemke to solve an LCP, contributed significantly to the development of the linear complementarity theory and brought the LCP into the limelight. Ever since, Lemke’s algorithm remains as an object of research for several decades. In this chapter, different variants of Lemke’s algorithm are discussed. Matrix classes play an important role for studying the theory of Lemke’s algorithm. Various matrix classes that are processable by Lemke’s algorithm are reviewed, and some related results are provided with examples to describe the various features of Lemke’s algorithm.