The subject of nonlinear partial differential equations has seen a lot of research on systems underlying basic theories in geometry, topology and physics. These mathematical models share the property of being derived from variational principles. Understanding the structure of critical configurations and the dynamics of the corresponding evolution problems is important for the development of physical theories and their applications. This volume contains survey lectures in four different areas, delivered at the 1995 Barrett Lectures held at the University of Tennessee. The lectures are on: nonlinear hyperbolic systems arising in field theory and relativity; harmonic maps from Minkowski spacetime; dynamics of vortices in the Ginzburg-Landau model of superconductivity; and the Seiberg-Witten equations and their application to problems in four-dimensional topology. the text should prove useful to graduate students and researchers in mathematical physics, partial differential equations, differential geometry and topology.