The Deep Underground Neutrino Experiment (DUNE) is an international project for neutrino physics and proton-decay searches, currently in the design and planning stages. Once built, DUNE will consist of two detectors exposed to the world's most intense neutrino beam. The near detector at Fermilab will record neutrino interactions near the beginning of the beam line. The other, much larger, detector, comprising four 10-kton liquid argon time projection chambers (LArTPCs), will be installed at a depth of 1.5 km at the Sanford Underground Research Facility in South Dakota, about 1,300 km away from the neutrino source. The unique combination of the high-intensity neutrino beam, DUNE's high-resolution near detector system, and massive LArTPC far detector enables a variety of probes of Beyond-the-Standard-Model (BSM) physics, from the discovery of new particles (sterile neutrinos or dark matter), to precision tests of beyond the three-flavor mixing paradigm, non-standard neutrino interactions (NSIs), heavy neutral leptons, and the detailed study of rare processes (e.g., neutrino trident production). This article reviews these physics topics and discusses the prospects for their discovery at the DUNE experiment.