Toomre's research and teaching centers on astrophysical fluid dynamics (AFD), with particular emphasis on nonlinear theories for compressible convection in stars, and in studying nonlinear dynamical systems exhibiting multiple bifurcations and chaos. He is particularly interested in topics where theoretical work related to fluid dynamics can be challenged and tested by observation and experiment. He is active in turbulence theory and simulation, with extensive experience in the use of vector supercomputers, massively parallel machines, high-speed networks, archival storage systems, and major visualization systems. Toomre's teaching spans graduate courses in basic fluid dynamics, waves and instabilities, nonlinear dynamical systems, geophysical and astrophysical fluid turbulence, magnetohydrodynamics, mathematical methods, computational mathematics and simulations, and a variety of courses in general astronomy.