In 1932, the great British fluid dynamicist Sir Horace Lamb wrote, "If I should come to heaven, then I want clarity about two things: quantum electrodynamics and turbulence. Where concerns the first desire, I am rather confident." Seventeen years later, we had a general theory of quantum electrodynamics; more than seventy years later, we still have no general theory of turbulence. Turbulence is difficult to understand because fluids are governed by nonlinear partial differential equations. The equations themselves have been known since the nineteenth century, but fundamental questions about them, such as the existence and uniqueness of their solutions, are still open. Numerical methods and computer simulation can shed some light on the behavior of solutions in the turbulent regime, but these require substantial supercomputer resources because of the coexistence of motion on many disparate length and time scales. My research focus is on theoretical and computational fluid dynamics, with emphasis on an improved understanding of fluid turbulence. My group has several fronts of attack on this problem: Computational fluid dynamics The dynamical systems approach to fluid dynamics Lattice models of fluid dynamics Topological fluid dynamics Vortex dynamics in two and three dimensions