Conversion of energy at the gas-solid interface lies at the heart of many industrial applications such as heterogeneous catalysis. Dissipation of parts of this energy into the substrate bulk drives the thermalization of surface species, but also constitutes a potentially unwanted loss channel. At present, little is known about the underlying microscopic dissipation mechanisms and their (relative) efficiency. At metal surfaces, prominent such mechanisms are the generation of substrate phonons and the electronically non-adiabatic excitation of electron-hole pairs. In recent years, dedicated surface science experiments at defined single-crystal surfaces and predictive-quality firstprinciples simulations have increasingly been used to analyze these dissipation mechanisms in prototypical surface dynamical processes such as gas-phase scattering and adsorption, diffusion, vibration, and surface reactions. In this topical review we provide an overview of modeling approaches to incorporate dissipation into corresponding dynamical simulations starting from coarsegrained effective theories to increasingly sophisticated methods. We illustrate these at the level of individual elementary processes through applications found in the literature, while specifically highlighting the persisting difficulty of gauging their performance based on experimentally accessible observables.