Liquid crystals generally support orientational singularities of the director
field known as topological defects. These latter modifiy transport properties
in their vicinity as if the geometry was non-Euclidean. We present a state of
the art of the differential geometry of nematic liquid crystals, with a special
emphasis on linear defects. We then discuss unexpected but deep connections
with cosmology and high-energy-physics, and conclude with a review on defect
engineering for transport phenomena