"Formulated by Einstein in 1916, General Relativity (GR) passed many stringent tests, and is now accepted as the standard theory of gravity and one of mankind's greatest achievements. Nevertheless, most experiments can only probe the low-curvature regime, while the strong curvature regime remains essentially unexplored. Exactly in this regime the dynamics of black holes (BHs) and neutron stars (NSs) can sensibly differ from GR, with potentially observable effects. BHs are among the most exciting predictions of many gravity theories and play a key role in fundamental physics. BHs and NSs share two unique properties: they have an enormous gravitational field and they are very common in the universe. Thus, they are perfect testbeds to probe strong-field effects in gravity.Strong motivations exist to go beyond GR. These include singularity and quantization issues, as well as astrophysical issues related to the dark matter and dark energy problem. Assuming GR as the correct theory of gravity in the elusive strong-field regime, we are introducing a bias, which can affect astrophysical observations and our understanding of the Universe.The most promising way to investigate strong curvature effects is by detecting gravitational waves (GWs) emitted from compact objects. Predicted by Einstein in 1916, GWs still have not been detected due to their feebleness. This state of affairs may change in the near future, thanks to the GW observatories operating in Europe, USA and Japan.Our ambitious program to test effects beyond GR in BH and NS physics may have a profound impact in astrophysics and in fundamental physics. In particular we aim to:- Develop a general framework, in which all viable corrections to GR can be accommodated and investigate effects beyond GR by means of ""GW spectroscopy""- Constrain alternative theories by a combined analysis of the GW spectrum from BHs and of the internal structure of NSs, using data-analysis techniques and state-of-the-art observations"