The geometric language of General Relativity is not normally related to Condensed Matter (CM) Physics since it is the electromagnetic and not the gravitational interaction that dominates the physics of CM systems. What points in common would then CMP have with Cosmology and the dynamics of objects in a gravitational field? There is at least one that is very important: topological defects formed in symmetry breaking phase transitions. To explore the similarities and differences here has been a very fruitful experience for both sides. On one hand, topological defects in solids started to be described by a gravity-like theory including torsion and, on the other hand, experiments have been proposed and performed in CM systems with the purpose of testing cosmological theories. Some examples are: 1) Landau levels and the Aharonov-Bohm effect of electrons moving in a crystal containing a screw dislocation can be described in a simple way in a geometric formalism; 2) closed timelike curves have been proposed in the vicinity of vortices in superfluid Helium; 3) Kibble mechanism, for the generation of topological defects, has been experimentally verified in liquid crystals. In summary, Condensed Matter Physics with its rich diversity of systems and phenomena and of relatively easy access to experiments, appears as a laboratory for testing hypotheses of gravitational theory and cosmology involving topological defects. In this work I summarize recent results in this interface area focusing mainly in the results obtained by our research group.