The Symposium on the Complexity of Computer Compu tations was held at the IBM Thomas J. Watson Research Center in Yorktown Heights, New York, March 20-22, 1972. These Proceedings contain all papers presented at the Symposium together with a transcript of the concluding panel discussion and a comprehensive bibliography of the field. The Symposium dealt with complexity studies closely re lated to how computations are actually performed on computers. Although this area of study has not yet found an appropriate or generally accepted name, the area is recognizable by the signif icant commonality in problems, approaches, and motivations. The area can be described and delineated by examples such as the following. (1) Determining lower bounds on the number of operations or steps required for computational solutions of specific problems such as matrix and polynomial calculations, sorting and other combinatorial problems, iterative com putations, solving equations, and computer resource allocation. (2) Developing improved algorithms for the solution of such problems which provide good upper bounds on the number of required operations, along with experimental and v vi PREFACE theoretical evidence concerning the efficiency and numer ical accuracy of those algorithms. (3) Studying the effects on the efficiency of computation brought about by variations in sequencing and the intro duction of parallelism.