Broad-based interest in microscale heat transport in novel materials, engineered phononic materials, metamaterials, and their relevant systems has created significant demand for computational approaches to aid in investigation and design of materials that support phonons. This review describes the significant improvements that have been made and new needs that have emerged for capabilities associated with the computability of phonons. The technical scope encompasses issues, especially relevant to bulk, interface, and surface effects. Traditional approaches such as molecular dynamics, lattice dynamics, and Boltzmann transport equation continue to advance the field but are frequently extended to the limits of their physical or numerical validity. New materials beyond traditional group-IV, III-V, and II-VI semiconductors, phenomena that critically depend on scattering, such as in low-dimensional nanostructures, materials with interior surfaces and defects, and in high-temperature environments, continue to push these limits. The basis for the traditional calculation methods shares their origins with the earliest theories for thermal transport, acoustic waves in solids, spectroscopy and dynamical crystal lattices. These will remain in wide use in the future. But computing methods and the accompanying advances in microprocessor technologies have enabled growth of phonon computing models and methods in sophistication, accuracy, fidelity and complexity that will lead to fundamental impacts beyond the classic types of problems for which they were developed. With their increasingly integrated use for design and research, the myriad developments that presently exist must be understood for their suitability for certain applications and their ability to aid in the pursuit of new technologies.