The amenability of the natural action of a discrete group on its Alexandroff (respectively on its Stone-Cech) compactification is equivalent to the amenability (respectively the exactness) of the group. Whereas amenability of groupoids has been widely studied, very few results are known concerning exactness of groupoids. Our purpose here is mainly to provide the topological background needed to investigate this notion. To that end, we study first the fibrewise compactifications of fibre spaces and in particular the Alexandroff and the Stone-Cech fibrewise compactifications. Starting from G-spaces, where G is an étale groupoid, we show that these two compactifications are G-spaces in a natural way. Applications to groupoids will be developed in a separate paper