This thesis discusses consequences of color-kinematics duality and applications towards computing quantum scattering amplitudes and classical radiation fields. Stemming from this duality, tree-level Bern-Carrasco-Johansson amplitude relations can be extended to one-loop integral coefficient relations for scattering in Yang-Mills theory. The double copy, which also follows from color-kinematics duality, allows for graviton scattering amplitudes to be found from scattering amplitudes in Yang-Mills theory. Additionally, a classical radiative double copy for obtaining gravitational waves in various theories is discussed. As a warm-up, a classical double copy of the Lienard-Wiechert potential in electrodynamics is found within a specific context, which allows for gravitational waves in linearized gravity to be found. Next, radiation in Yang-Mills and Yang-Mills-biadjoint-scalar theories is found, and the radiative double copy of these results allows for radiation in general relativity and Einstein-Yang-Mills theory, repsectively. In light of the recent detection of gravitational waves by the LIGO collaboration, this motivates the search of efficient analytic techniques for computing gravitational radiation. The double copy offers a way to apply methods from particle physics to gravitational-wave astronomy.