Single-molecule spectroscopy is a powerful method for studying the physics of molecular systems, particularly biomolecules, such as proteins and nucleic acids. By avoiding ensemble averaging, single-molecule techniques can resolve structural distributions and fluctuations even for complex and conformationally heterogeneous systems; they also reveal the close link between biological function and the statistical mechanics of the underlying processes. The combination of single-molecule fluorescence detection with Förster resonance energy transfer has become an essential tool for probing biomolecular dynamics on timescales ranging from nanoseconds to days. This Review briefly outlines the state of the art of single-molecule Förster resonance energy transfer spectroscopy and then highlights some of the most important physics-based developments that are expected to further expand the scope of the technique. Key areas of progress include improved time resolution, access to nonequilibrium dynamics and synergies with advances in data analysis and simulations. These developments create new opportunities for attaining a comprehensive understanding of the dynamics and functional mechanisms of biological processes at the nanoscale.