Black holes and the cosmos itself are among the most mysterious and fascinating objects in the Universe. Gravitational waves from compact binary coalescences can shed light on the nature of both these phenomena as they are perfect probes of both their sources and the space-time through which they propagate. In this thesis, we investigate both: Part I deals with the inference of the properties of space-time close to black hole remnants of the merger of binary black holes observed by the LIGO-Virgo interferometers; Part II instead concentrates on the inference of the dynamics of the Universe from the observation of extreme mass-ratio inspirals that will be observed by the Laser Interferometer Space Antenna at cosmological distances.