  General relativity does not allow one to specify the topology of space,
leaving the possibility that space is multi-- rather than simply--connected. We
review the main mathematical properties of multi--connected spaces, and the
different tools to classify them and to analyse their properties. Following the
mathematical classification, we describe the different possible muticonnected
spaces which may be used to construct universe models. We briefly discuss some
implications of multi--connectedness for quantum cosmology, and its
consequences concerning quantum field theory in the early universe. We consider
in details the properties of the cosmological models where space is
multi--connected, with emphasis towards observable effects. We then review the
analyses of observational results obtained in this context, to search for a
possible signature of multi--connectedness, or to constrain the models. They
may concern the distribution of images of cosmic objects like galaxies,
clusters, quasars,..., or more global effects, mainly those concerning the
Cosmic Microwave Background, and the present limits resulting from them.
