Gravitational-wave (GW) signals from the coalescence of almost a hundred binary systems have been detected over the past few years. These observations have improved our understanding of binary black holes and neutron stars, their properties, and astrophysical formation channels. GWs also probe gravity in the nonlinear, strong-field regime, thus allowing us to search for, or constrain, deviations from general relativity.

The focus of this dissertation is improving the analytical description of binary dynamics, which is important for producing accurate waveform models that can be used in searching for GW signals, inferring their parameters, and testing gravity. The research presented here can be divided into three complementary parts: 1) extending the post-Newtonian (PN) approximation for spinning binaries to higher orders, 2) improving effective-one-body (EOB) waveform models, and 3) identifying some signatures of modified gravity theories in waveforms.

The PN approximation, valid for slow motion and weak gravitational field, is widely used to model the dynamics of comparable-mass binaries, which are the main GW sources for ground-based detectors. We derive PN results for spinning binaries at the third- and fourth-subleading PN orders for the spin-orbit coupling, and at the third-subleading order for the spin(1)-spin(2) coupling. We adopt an approach that combines several analytical approximation methods to obtain PN results valid for arbitrary mass ratios from gravitational self-force results at first order in the mass ratio. This is possible due to the simple mass dependence of the scattering angle in the post-Minkowskian approximation (weak field but arbitrary velocities).

The EOB formalism produces accurate waveforms by combining analytical results for the binary dynamics with numerical relativity information, while recovering the strong-field test-body limit. To improve EOB models, we include spin-precession effects in the Hamiltonian up to the fourth PN order, and extend the radiation-reaction force and waveform to eccentric orbits. We also assess the accuracy of post-Minkowskian results, for both bound and scattering orbits, and incorporate them in EOB Hamiltonians.

In the context of modified gravity theories, we derive the conservative and dissipative dynamics in Einstein-Maxwell-dilaton theory at the next-to-leading PN order, and compute the Fourier-domain gravitational waveform. We also develop a theory-agnostic effective-field-theory approach for describing spontaneous and dynamical scalarization: non-perturbative phenomena in which compact objects can undergo a phase transition and acquire scalar charge. We apply this approach to binary black holes in Einstein-Maxwell-scalar theory using a quasi-stationary approximation, then extend it to account for the dynamical evolution of the scalar charge, and apply it to binary neutron stars in a class of scalar-tensor theories.

Improving waveform models is important for current-generation GW detectors and necessary for future detectors, such as LISA, the Einstein telescope, and Cosmic Explorer. The results obtained in this work are important steps towards that goal.