  General relativity does not allow one to specify the topology of space,
leaving the possibility that space is multiply rather than simply connected. We
review the main mathematical properties of multiply connected spaces, and the
different tools to classify them and to analyse their properties. Following
their mathematical classification, we describe the different possible
muticonnected spaces which may be used to construct Friedmann-Lemaitre universe
models. Observational tests concern the distribution of images of discrete
cosmic objects or more global effects, mainly those concerning the Cosmic
Microwave Background. According to the 2003-2006 WMAP data releases, various
deviations from the flat infinite universe model predictions hint at a possible
non-trivial topology for the shape of space. In particular, a finite universe
with the topology of the Poincar\'e dodecahedral spherical space fits
remarkably well the data and is a good candidate for explaining both the local
curvature of space and the large angle anomalies in the temperature power
spectrum. Such a model of a small universe, whose volume would represent only
about 80% the volume of the observable universe, offers an observational
signature in the form of a predictable topological lens effect on one hand, and
rises new issues on the physics of the early universe on the other hand.
