This thesis is concerned with topological groupoids, that is, with categories in which each morphism is an isomorphism topologised in such a way that the algebraic operations are compatible with the topology. Three main areas are examined, they are: topological aspects, measure theoretic considerations and, thirdly, representations of groupoids. In the first of these, it is shown that the base space of the universal bundle of J. Milnor is a classifying space for certain topological groupoids. The second aspect concerns a notion of invariant measure for groupoids which generalises that of a group.
It is shown that such measures always exist on a locally compact Hausdorff topological groupoid, and a classification is given with suitable restrictions. Convolution-algebras are then constructed and applications to differential geometry and transformation groups are considered. Finally, a theory of unitary representations of locally compact Hausdorff topological groupoids is presented. Amongst the results obtained, is a version for groupoids of the classical Peter-Weyl theory for groups.