Topology of the Universe: Theory and Observation

``One could imagine that as a result of enormously extended astronomical experience, the entire universe consists of countless identical copies of our Milky Way, that the infinite space can be partitioned into cubes each containing an exactly identical copy of our Milky Way. Would we really cling on to the assumption of infinitely many identical repetitions of the same world? ... We would be much happier with the view that these repetitions are illusory, that in reality space has peculiar connection properties so that if we leave any one cube through a side, then we immediately reenter it through the opposite side.'' (Schwarzschild 1900, translation 1998) Developments in the theoretical and observational sides of cosmic topology were slow for most of the century, but are now progressing rapidly, at the scale of most interest which is 1-10 h^-1 Gpc rather than 10kpc. The historical, mathematical and observational sides of this subject are briefly reviewed in this course.