RESUMEN

I review recent progress in the understanding of certain scenarios for the Planck-scale structure of spacetime, with emphasis on the possibility that Lorentz symmetry may be broken of deformed. Especially the idea of deformed Lorentz symmetry, in the sense of the so-called Dounbly Special Relativity, must still be investigated on the conceptual side in order to establish some of its core physical implications, but evidence in support of the emergence of the deformation in noncommutative geometry and (perhaps) Loop Quantum Gravity is growing. The natural arena for an associated phenomenology is in the astrophysics of gamma rays and cosmic rays, and forthcoming data may provide valuable constraints and/or insight.