Resumen : It is an understatement that symmetries play a central role in physics. Symmetries are in fact omnipresent in the formulation of all fundamental physical laws. The talk will provide a pedagogical review of some aspects of symmetry. The emphasis will be put on recent developments aimed at reaching a more fundamental understanding of the gravitational force, where the somewhat mysterious "dualities" (a special kind of symmetry) appear to be the tip of the iceberg of a remarkable, yet-to-be-discovered structure. These recent developments, in which symmetry meets infinity, are at the frontiers of current mathematical and physics research.