This paper unifies the multipole formalisms for gravitational radiation constructed by various researchers and extends their results. It reviews scalar, vector, and tensor spherical harmonics used in general relativity, including Regge-Wheeler harmonics, symmetric trace-free (STF) tensors, Newman-Penrose spin-weighted harmonics, and Mathews-Zerilli-Clebsch-Gordan-coupled harmonics. The paper introduces the concept of a local wave zone to separate wave generation from wave propagation. It decomposes the generic radiation field in the local wave zone into multipole components and expresses the energy, linear momentum, and angular momentum in the waves as infinite sums of contributions. For slow-motion sources admitting a nonsingular, spacetime-covering de Donder coordinate system, the multipole moments of the radiation field are expressed as volume integrals over the source. These source integrals are reexpressed as infinite power series in the ratio of the source size to the reduced wavelength of the waves, leading to Newtonian, post-Newtonian, and post-post-Newtonian formulas. The paper also derives a multipole-moment wave-generation formalism for slow-motion sources with arbitrarily strong internal gravity, including systems that cannot be covered by de Donder coordinates. This formalism involves calculating the source's instantaneous, near-zone, external gravitational field as a solution of the time-independent Einstein field equations and reading off the source's instantaneous multipole moments from this field. The paper concludes with a detailed presentation of the new slow-motion wave-generation formalism.This paper unifies the multipole formalisms for gravitational radiation constructed by various researchers and extends their results. It reviews scalar, vector, and tensor spherical harmonics used in general relativity, including Regge-Wheeler harmonics, symmetric trace-free (STF) tensors, Newman-Penrose spin-weighted harmonics, and Mathews-Zerilli-Clebsch-Gordan-coupled harmonics. The paper introduces the concept of a local wave zone to separate wave generation from wave propagation. It decomposes the generic radiation field in the local wave zone into multipole components and expresses the energy, linear momentum, and angular momentum in the waves as infinite sums of contributions. For slow-motion sources admitting a nonsingular, spacetime-covering de Donder coordinate system, the multipole moments of the radiation field are expressed as volume integrals over the source. These source integrals are reexpressed as infinite power series in the ratio of the source size to the reduced wavelength of the waves, leading to Newtonian, post-Newtonian, and post-post-Newtonian formulas. The paper also derives a multipole-moment wave-generation formalism for slow-motion sources with arbitrarily strong internal gravity, including systems that cannot be covered by de Donder coordinates. This formalism involves calculating the source's instantaneous, near-zone, external gravitational field as a solution of the time-independent Einstein field equations and reading off the source's instantaneous multipole moments from this field. The paper concludes with a detailed presentation of the new slow-motion wave-generation formalism.