Structural complexity theory is the study of the form and meaning of computational complexity classes. Complexity classes - P, NP, Probabilistic P, PSPACE, etc. - are formalizations of computational powers - deterministic, nondeterministic, probabilistic, etc. By examining the structure of and the relationships between these classes, we seek to understand the relative strengths of their underlying computational paradigms. This thesis studies counting in structural complexity theory. We are interested in complexity classes defined by counting and in the use of counting to explore the structure of these and other classes.