In this dissertation I explore several topics in the field of gravitational wave astronomy. By means of introduction, I review the historical evolution of humanity's understanding of the mechanics of gravity, and the events which eventually led to the first ever detection of gravitational waves in 2015.

The first half of the thesis is dedicated to the effect which gravitational waves have on the apparent position of stars on the sky. The astrometric shift caused by a gravitational wave signal can be quantified, and precise astrometric measurements (from $Gaia$) can provide a new method for searching for low-frequency GWs. This method is applied to searches for signals from individually resolvable supermassive black hole binaries. The main obstacle to performing efficient searches is the large size of the data sets, which consist of more than one billion stars. A near-lossless compression which reduces the size of the data set by a factor of $106$ is discussed and implemented. Mock data sets are generated to simulate detections of gravitational waves using this method, and the frequency and directional sensitivities of the full-term $Gaia$ mission are calculated. Parallels are drawn with the field of pulsar timing searches for GWs. This knowledge of the astrometric response is used to address the problem of searching for low frequency gravitational wave backgrounds using astrometric measurements. The astrometric deflections due to a stochastic GW background form a correlated vector field on the sphere (sky). Using a convenient decomposition of the correlation matrix, the 2-point correlation functions are calculated and compared to the redshift correlation in pulsar timing literature (and the Hellings-Downs curve). The correlation between redshift and astrometric deflections is also considered.

The second part of the dissertation focuses on the problem of resonances in extreme mass-ratio in-spirals (EMRIs). These events are prime candidates for GW detection in the millihertz band (by detectors like LISA), and involve a stellar mass black hole (or a similar compact object) merging with a supermassive black hole. Properties of the trajectory of the lighter body are well known, however little is known about the behaviour of such systems during resonance of the radial and polar motions. Two existing models for this behaviour are described: the instantaneous frequency approach (developed by Gair, Bender, and Yunes) and the two timescales approach (proposed by Flanagan and Hinderer). Both methods depend on exact treatment of the gravitational self-force, which is currently not available. The results of Gair, Bender, and Yunes are extended to higher-order in the on-resonance flux modification, and the instantaneous frequency approach is confirmed to be a valid treatment of this problem. The algorithm for finding higher-order solutions is described, and further directions for extending this research are proposed.