  Liquid crystals are assemblies of rod-like molecules which self-organize to
form mesophases, in-between ordinary liquids and anisotropic crystals. At each
point, the molecules collectively orient themselves along a privileged
direction, which locally defines an orientational order. Sometimes, this order
is broken and singularities appear in the form of topological defects. This
tutorial article is dedicated to the geometry, topology and physics of these
defects. We introduce the main models used to describe the nematic phase and
discuss the isotropic-nematic phase transition. Then, we present the different
families of defects in nematics and examine some of their physical outcomes.
Finally, we show that topological defects are universal patterns of nature,
appearing not only in soft matter, but also in biology, cosmology, geology and
even particle physics.
