The discovery that the Universe is accelerating will possibly be seen in a few years as one of the turning points in the history of cosmology. After nearly ten years, the origin of the accelerated expansion is still unknown. One possibility is that the Universe is permeated by a "dark energy" which gives rise to a gravitational repulsion. Alternatively, perhaps the very equations of General Relativity need to be modified or generalized to higher-dimensional worlds. In my talk I will first review these general ideas. I will then introduce a new promising approach to measuring the properties of dark energy from large surveys of galaxy redshifts, presenting a first application to available data. Specifically, I will show how measurements of linear redshift-space distortions due to galaxy coherent motions can be used to trace the evolution of the growth rate of cosmic structures f(z) at different epochs. Combined to the classical estimates of the cosmic expansion rate H(z), as those from Type Ia Supernovae or Baryonic Acoustic Oscillations, these observations can pinpoint the physical origin of the accelerated expansion, distinguishing true ?dark-energy? models from modifications of the gravitation theory.