New precision observations of compact objects and the imminent advent of gravitational-wave (GW) astronomy provide us with the unique opportunity to test fundamental physics with astrophysical observations to unprecedented level. Neutron stars (NSs) and black holes (BHs) can be used as cosmic labs where matter in extreme conditions, particle physics, and even the very foundations of Einstein's theory can be put to the test. Triggered by recent breakthroughs at various levels, the area of strong gravity is experiencing a second Golden Age. In parallel with novel electromagnetic observations, advanced GW observatories in Europe and USA will open new windows to the unexplored strong-gravity regime and will finally shed light on the properties of ultradense matter in NS cores. The potential of GW astrophysics is enormous and far to be fully explored. Counterintuitive effects taking place near isolated compact objects have been recently discovered, but their GW signatures in realistic environment remain to be investigated. We are now in the exciting position of using observations to make contact between relativistic astrophysics and fundamental questions. The goal of our innovative project is to connect this missing link. We propose to investigate strong-gravity effects via precision GW phenomenology. In particular we aim to: 1) Develop semianalytical methods to study NS-NS binaries and spinning isolated NSs, and to constrain the behavior of matter at nuclear density using GW observations; 2) Develop a model-independent framework to study GW signatures of accretion onto massive BHs; 3) Investigate the interplay between “BH bomb” instabilities and accretion in the context of puzzling phenomena, such as jet emission or gamma-ray bursts; 4) Constrain dark matter candidates by studying their interaction with BHs and NSs in realistic scenarios. Our proposal is located at the interface between astrophysics and fundamental physics and can have a profound impact for both.