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