  We propose a unified framework in which the different constructions of
cohomology groups for topological and Lie groups can all be treated on equal
footings. In particular, we show that the cohomology of "locally continuous"
cochains (respectively "locally smooth" in the case of Lie groups) fits into
this framework, which provides an easily accessible cocycle model for
topological and Lie group cohomology. We illustrate the use of this unified
framework and the relation between the different models in various
applications. This includes the construction of cohomology classes
characterizing the string group and a direct connection to Lie algebra
cohomology.
