This thesis concerns the use of observations of gravitational waves as tools for astronomy and fundamental physics. Gravitational waves are small ripples in spacetime produced by rapidly accelerating masses; their existence has been predicted for almost 100 years, but the first direct evidence of their existence came only very recently with the announcement in February 2016 of the detection by the LIGO and VIRGO collaborations. Part I of this thesis presents an introduction to gravitational wave astronomy, including a detailed discussion of a wide range of gravitational wave sources, their signal morphologies, and the experimental detectors used to observe them. Part II of this thesis concerns a particular data analysis problem which often arises when trying to infer the source properties from a gravitational wave observation. The use of an inaccurate signal model can cause significant systematic errors in the inferred source parameters. The work in this section concerns a proposed technique, called the Gaussian process marginalised likelihood, for overcoming this problem. Part III of this thesis concerns the possibility of testing if the gravitational field around an astrophysical black hole conforms to the predictions of general relativity and the cosmic censorship hypothesis. It is expected that the gravitational field should be well described by the famous Kerr solution. Two approaches for testing this hypothesis are considered; one using X-ray observations and one using gravitational waves. The results from these two approaches are compared and contrasted. Finally, the conclusions and a discussion of future prospects are presented in part IV of this thesis.