Astronomy with gravitational-wave observations is now a reality. Much of the theoretical research in this field falls under three broad themes: the mathematical description and physical understanding of gravitational radiation and its effects; the construction of accurate and computationally efficient waveform models for astrophysical sources; and the improved statistical analysis of noisy data from interferometric detectors, so as to extract and characterise source signals. The doctoral thesis presented in this dissertation is an investigation of various topics across these themes.

Under the first theme, we examine the direct interaction between gravitational waves and electromagnetic fields in a self-contained theoretical study; this is done with a view to understanding the observational implications for highly energetic astrophysical events that radiate in both the gravitational and electromagnetic sectors. We then delve into the second theme of source modelling by developing and implementing an improved waveform model for the extreme-mass-ratio inspirals of stellar-mass compact objects into supermassive black holes, which are an important class of source for future space-based detectors such as the Laser Interferometer Space Antenna.

Two separate topics are explored under the third theme of data analysis. We begin with the procedure of searching for gravitational-wave signals in detector data, and propose several combinatorial compression schemes for the large banks of waveform templates that are matched against putative signals, before studying the usefulness of these schemes for accelerating searches. After a gravitational-wave source is detected, the follow-up process is to measure its parameters in detail from the data; this is addressed as we apply the machine-learning technique of Gaussian process regression to gravitational-wave data analysis, and in particular to the formidable problem of parameter estimation for extreme-mass-ratio inspirals.