This chapter provides an overview of the development and applications of homotopy invariant algebraic structures on topological spaces. The authors, J. M. Boardman and R. M. Vogt, begin by discussing the motivation and historical context, highlighting the use of categories of operators (PROPs) to identify topological spaces with algebraic structures through functors. They trace the origins of this approach to a seminar by Stasneff in 1967, where the idea of a topological analogue of PACTs was introduced. The chapter then delves into the theoretical framework, covering topics such as multi-colored theories, the bar construction for theories and PROPs, and the development of homotopy invariant structures. It introduces the concept of E-structures, which are crucial for characterizing infinite loop spaces. The authors compare their method with other approaches, including those by Segal, Beck, and May, emphasizing the advantages of their approach in terms of generality and the ability to handle more complex algebraic structures. The chapter also includes detailed treatments of n-fold and infinite loop spaces, Milgram's classifying space construction, and the recognition principle for these spaces. It concludes with a discussion on the broader applications of homotopy colimits in homotopy theory. The authors acknowledge the contributions of various individuals and institutions that supported their research.This chapter provides an overview of the development and applications of homotopy invariant algebraic structures on topological spaces. The authors, J. M. Boardman and R. M. Vogt, begin by discussing the motivation and historical context, highlighting the use of categories of operators (PROPs) to identify topological spaces with algebraic structures through functors. They trace the origins of this approach to a seminar by Stasneff in 1967, where the idea of a topological analogue of PACTs was introduced. The chapter then delves into the theoretical framework, covering topics such as multi-colored theories, the bar construction for theories and PROPs, and the development of homotopy invariant structures. It introduces the concept of E-structures, which are crucial for characterizing infinite loop spaces. The authors compare their method with other approaches, including those by Segal, Beck, and May, emphasizing the advantages of their approach in terms of generality and the ability to handle more complex algebraic structures. The chapter also includes detailed treatments of n-fold and infinite loop spaces, Milgram's classifying space construction, and the recognition principle for these spaces. It concludes with a discussion on the broader applications of homotopy colimits in homotopy theory. The authors acknowledge the contributions of various individuals and institutions that supported their research.