  A general framework for integration over certain infinite dimensional spaces
is first developed using projective limits of a projective family of compact
Hausdorff spaces. The procedure is then applied to gauge theories to carry out
integration over the non-linear, infinite dimensional spaces of connections
modulo gauge transformations. This method of evaluating functional integrals
can be used either in the Euclidean path integral approach or the Lorentzian
canonical approach. A number of measures discussed are diffeomorphism invariant
and therefore of interest to (the connection dynamics version of) quantum
general relativity. The account is pedagogical; in particular prior knowledge
of projective techniques is not assumed. (For the special JMP issue on
Functional Integration, edited by C. DeWitt-Morette.)
