An advanced monograph on the subject of topological transformation groups, this volume summarizes important research conducted during a period of lively activity in this area of mathematics. The book is of particular note because it represents the culmination of the authors' research, in collaboration with Andrew Gleeson of Harvard University, which led to their solution of a well-known mathematical conjecture, Hilbert's Fifth Problem. AUTHORS: Deane Montgomery (1909 92) received his Ph.D. from the University of Iowa, taught at Smith College from 1935 46, and was at the Institute for Advanced Study in Princeton from 1947 until his retirement in 1980. He was President of the American Mathematical Society in 1961 62. Leo Zippin (1905 95) received his Ph.D. from the University of Pennsylvania. He taught at Queens College in New York from 1938 until his retirement in 1971.