In this thesis we deal with questions of continuous group cohomology of continuous representations of a separable locally compact group on a real or complex Banach space. Of particular importance is the case of a compact group. Here we use a?ne actions to prove vanishing theorems. To do this, we give an alternative de?nition of the cohomology, which is recursive. As a consequence we prove under certain conditions (equivalent with the existence of a non-trivial simultaneous ?xed point of the associated a?ne map) all cohomology groups vanish. When G is a connected Lie group, we study the relationship of its cohomology with the corresponding Lie algebra cohomology. Finally, we consider the situation of a closed subgroup H of G which is cocompact and of co?nite volume and show just as in the case of a compact group that the restriction map H^n(G,V)-->H^n(H,V) is injective and apply this to questions of complete reducibility of representations.