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Garrett Birkhoff’s mathematical work over the last three decades has been primarily centered on scientific computation and numerical analysis, and the other contributions to this special issue survey the broad range of his research and influence in this area. However, this subject represents only a small part of Garrett’s many and diverse interests, and the number of areas of mathematics on which he has left his mark is astonishing. My own contribution to these proceedings will be devoted to another of Garrett’s “applied” research areas-tile applications of Lie groups to differential equations, a subject whose renaissance, especially among applied mathematicians, owes much to his pioneering efforts in the 194Os, [2j. In light of the subject of this conference, I find Garrett’s choice of thesis topic for his last pure mathematics Ph.D. student, [15], slightly ironic, but his insight into its hidden potentialities has more than been fulfilled. My debt to Garrett for initiating my research career and its subsequent course cannot be understated. So it is with great pleasure that I am able to contribute to these proceedings a discussion of some of my own investigations into the many uses of symmetry for studying differential equations. More specifically, this survey will cover recent (and not so recent) developments in the use of symmetry and other methods for finding explicit solutions to systems of partial differential equations. While cstensibly not directly connected with the general theme of scientific computing, this particular topic does have several points of contact with the other papers in this volume. First, and perhaps of crucial importance, is the role that explicit solutions to partial differential equations play in the design and testing of numerical integrators. Being able to reproduce an explicit soiution to a complicated system of partial differential equations constitutes a reassuring check for the accuracy and reliability of one’s numerical package; moreover, explicit solutions can effectively serve as benchmarks for comparison of competing packages, or in establishing practical error estimates (which might be far more reasonable than any resulting from a rigorous theoretical analysis, if such is even available). Secondly, there is a well-stablished role played by computer algebra (symbolic manipulation) systems, such as MACSYMA, REDUCE, MAPLE and MATHEMATICA, in the determination of symmetry groups and