Fluids with suspended microstructure - complex fluids - are common actors in micro- and biofluidics applications and can have fascinating dynamical behaviors. A new area of complex fluid dynamics concerns "active fluids" which are internally driven by having dynamic microstructure such as swimming bacteria. Such motile suspensions are important to biology, and are candidate systems for tasks such as microfluidic mixing and pumping. To understand these systems, we have developed both first-principles particle and continuum kinetic models for studying the collective dynamics of hydrodynamically interacting microswimmers. The kinetic model couples together the dynamics of a Stokesian fluid with that of an evolving "active" stress field. It has a very interesting analytical and dynamical structure, and predicts critical conditions for the emergence of hydrodynamic instabilities and fluid mixing. These predictions are verified in our detailed particle simulations, and are consistent with current experimental observation.