Redshift surveys of galaxies beyond the local Universe (z ≫ 0.1) are opening up new possibilities to understanding the observed acceleration of cosmic expansion, one of the greatest mysteries of modern science. Baryonic Acoustic Oscillations in the galaxy power spectrum (or correlation function), provide us with a standard rod to measure the expansion history H(z). At the same time, redshift-space distortions in the clustering pattern due to galaxy peculiar motions are a measure of the growth rate of structure f(z). The combination of these two quantities, allows us to distinguish whether cosmic acceleration is due to the existence of a "dark energy" in the cosmic budget, or rather requires a modification of General Relativity. These two radically alternative scenarios are degenerate when considering H(z) alone, as yielded, e.g. by the Hubble diagram of Type Ia supernovae. In this short review paper I will mostly concentrate on the latter measurement, whose potential importance in this context has been recently highlighted. Current results are consistent with the simplest GR-based cosmological constant scenario, but error bars are still large. Detailed forecasts show that next-generation deep surveys optimizing the combination of large volumes and good galaxy sampling will be able to use redshift distortions as a key tool to understand the physical origin of cosmic acceleration. Among these, I introduce the newly started VIMOS Public Extragalactic Redshift Survey (VIPERS) at the ESO VLT, which is building at [Formula: see text] a sample comparable to the local 2dFGRS. Expectations from even larger surveys planned from space-borne observatories such as EUCLID will also be mentioned.