Background -- Algebraic Monoids -- Regularity Conditions -- Classification of Reductive Monoids -- Universal Constructions -- Orbit Structure of Reductive Monoids -- The Analogue of the Bruhat Decomposition -- Representations and Blocks of Algebraic Monoids -- Monoids of Lie Type -- Cellular Decomposition of Algebraic Monoids -- Conjugacy Classes -- The Centralizer of a Semisimple Element -- Combinatorics Related to Algebraic Monoids -- Survey of Related Developments.