Experiments and simulations indicate that suspensions of swimming microorganisms can exhibit complex phenomena, including pattern-forming instabilities, large scale fluid motions and enhanced passive scalar transport. This talk is an overview of theoretical and computational work describing some of these phenomena. Emphasis will be placed on analysis of various correlation functions associated with the dynamics. The talk concludes with a brief presentation of new directions in this area, including the coordinated motion of collections of bacterial flagella. This is joint work with Patrick T. Underhill and Pieter J. A. Jansssen.