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In the last decade, the study of the overall shape of the universe, calledCosmic Topology, has become testable by astronomical observations, especiallythe data from the Cosmic Microwave Background (hereafter CMB) obtained by WMAPand Planck telescopes. Cosmic Topology involves both global topologicalfeatures and more local geometrical properties such as curvature. It deals withquestions such as whether space is finite or infinite, simply-connected ormulti-connected, and smaller or greater than its observable counterpart. Astriking feature of some relativistic, multi-connected small universe models isto create multiples images of faraway cosmic sources. While the last CMB(Planck) data fit well the simplest model of a zero-curvature, infinite spacemodel, they remain consistent with more complex shapes such as the sphericalPoincare Dodecahedral Space, the flat hypertorus or the hyperbolic Picard horn.We review the theoretical and observational status of the field.

The Status of Cosmic Topology after Planck Data