This dissertation covers three topics in general relativity, each linked to notions of angular momentum. I first present a review of conservation laws in general relativity, discussing those associated with point-particle motion, field theories on a fixed background, and asymptotic symmetries for theories with no fixed background. This is followed by a review of the physics of spinning black holes, which are described by the Kerr spacetime. I lastly provide a review of gravitational waves, focusing on two physical phenomena that are relevant to this dissertation: extreme mass-ratio inspirals, an important source for future space-based gravitational wave detectors, and the gravitational wave memory effect, a permanent change in the separation of two freely-falling bodies caused by a burst of gravitational waves. The first of the three topics covered in this dissertation concerns the Carter constant, a generalization of the notion of total angular momentum for point particles in the Kerr spacetime. The Carter constant is of great interest, as understanding how it evolves for an inspiralling particle is important for determining the gravitational wave signal. I first consider generalizations of the Carter constant to general field theories in the Kerr spacetime, and show that no generalizations of the Carter constant that depend only on the stress-energy tensor of the theory can be conserved. However, this result does not eliminate the possibility that such a generalization, not constructed from the stress-energy tensor, can exist in particular field theories. I next discuss how one can construct conserved currents in linearized gravity on a Kerr background which generalize the Carter constant. These currents generalize the Carter constant in the following sense: in the geometric optics limit, they are related to the Carter constants of individual gravitons. For the second topic, I discuss generalizations of the gravitational wave memory effect. These generalizations, called "persistent gravitational wave observables", measure enduring effects following a burst of gravitational waves. This dissertation contains three examples of such persistent observables, as well as general techniques to calculate them, both in general spacetimes and in exact plane gravitational wave spacetimes. The first example of a persistent observable is a generalization of geodesic deviation that allows for arbitrary acceleration. The second example is a holonomy around a closed loop in spacetime of a connection related to linear and angular momentum. Finally, the third example is an explicit procedure by which an observer could measure persistent effects using a spinning test particle. The final topic considered is prescriptions for defining asymptotic charges in theories with no fixed background, and in particular angular momentum in Einstein-Maxwell theory. This is motivated by a strange result in electromagnetism, that the flux of angular momentum through null infinity, computed using the stress-energy tensor, depends on both radiative and Coulombic degrees of freedom. I first show that this situation carries over to electromagnetism on non-dynamical, asymptotically flat spacetimes for fluxes associated with the Lorentz symmetries in the asymptotic Bondi-Metzner-Sachs algebra. I then consider asymptotic charges (such as mass and angular momentum) in Einstein-Maxwell theory, where the metric is now dynamical. One could define these charges by using the same expressions as in vacuum general relativity, but such a prescription results in fluxes for the Lorentz charges that depend on Coulombic degrees of freedom, much as in the non-dynamical case. The correct approach is to use the prescription of Wald and Zoupas to compute the charges associated with any asymptotic symmetry on cross-sections of null infinity. The flux of this "new" notion of angular momentum depends only on radiative degrees of freedom and not Coulombic ones.