List of Figures

	Figure 1: Detector coordinate system and gravitational-wave coordinate system.
	Figure 2: Antenna pattern response functions of an interferometer (see Eqs. (58 *)) for $ψ = 0$ . Panels (a) and (b) show the plus ( $\|F+ \|$ ) and cross ( $\|F ×\|$ ) modes, panels (c) and (d) the vector x and vector y modes ( $\|F \| x$ and $\|F \| y$ ), and panel (e) shows the scalar modes (up to a sign, it is the same for both breathing and longitudinal). Color indicates the strength of the response with red being the strongest and blue being the weakest. The black lines near the center give the orientation of the interferometer arms.
	Figure 3: Antenna patterns for the pulsar-Earth system. The plus mode is shown in (a), breathing modes in (b), the vector-x mode in (c), and longitudinal modes in (d), as computed from Eq. (75 *). The cross mode and the vector-y mode are rotated versions of the plus mode and the vector-x mode, respectively, so we did not include them here. The gravitational wave propagates in the positive $z$ -direction with the Earth at the origin, and the antenna pattern depends on the pulsar’s location. The color indicates the strength of the response, red being the largest and blue the smallest.
	Figure 4: Schematic diagram of the projection of the data stream $d$ orthogonal to the GR subspace spanned by $F +$ and $F ×$ , along with a perpendicular subspace, for 3 detectors to build the GR null stream.
	Figure 5: Top: Fitting curves (solid curve) and numerical results (points) of the universal I-Love (left) and Q-Love (right) relations for various equations of state, normalized as $I¯= I ∕M 3NS$ , $¯λ (tid) = λ(tid)∕M 5NS$ and $Q¯ = − Q(rot)∕[M 3NS(S∕M N2S )2]$ , $MNS$ is the neutron-star mass, $λ(tid)$ is the tidal Love number, $Q(rot)$ is the rotation-induced quadrupole moment, and $S$ is the magnitude of the neutron-star spin angular momentum. The neutron-star central density is the parameter varied along each curve, or equivalently the neutron-star compactness. The top axis shows the neutron star mass for the APR equation of state, with the vertical dashed line showing $MNS = 1M ⊙$ . Bottom: Relative fractional errors between the fitting curve and the numerical results. Observe that these relations are essentially independent of the equation of state, with loss of universality at the 1% level. Image reproduced by permission from [452], copyright by APS.

	Abstract
1	Introduction
	1.1	The importance of testing
	1.2	Testing general relativity versus testing alternative theories
	1.3	Gravitational-wave tests versus other tests of general relativity
	1.4	Ground-based vs space-based detectors and interferometers vs pulsar timing
	1.5	Notation and conventions
2	Alternative Theories of Gravity
	2.1	Desirable theoretical properties
	2.2	Well-posedness and effective theories
	2.3	Explored theories
	2.4	Currently unexplored theories in the gravitational-wave sector
3	Detectors
	3.1	Gravitational-wave interferometers
	3.2	Pulsar timing arrays
4	Testing Techniques
	4.1	Coalescence analysis
	4.2	Burst analyses
	4.3	Stochastic background searches
5	Compact Binary Tests
	5.1	Direct and generic tests
	5.2	Direct tests
	5.3	Generic tests
	5.4	Tests of the no-hair theorems
6	Musings About the Future
	Acknowledgements
	References
	Footnotes
	Figures
	Tables