This is the ACM 1985 Turing Award Lecture paper. In this expository paper, the author traces the history of the work in the area of computational complexity. The chart on the development of combinatorial optimization and computational complexity very succinctly presents all the major developments from 1900 to the current time. The author has outlined the work of various people, such as M. Held, L. Ford, and S. Cook. By its very nature, the paper does not get into any technical details. However, the account of work done in the area of the Traveling Salesman problem and other NP-complete problems is very enlightening. The account of the work on linear programming problems and application of probability techniques to simplex algorithms is fairly detailed. This paper has no bibliography. Nonetheless, the paper should appeal to a very wide audience.