We discuss novel ways in which neutrino oscillation experiments can probe dark matter. In particular, we focus on interactions between neutrinos and ultralight (“fuzzy”) dark matter particles with masses of order 10^-22 eV. It has been shown previously that such dark matter candidates are phenomenologically successful and might help ameliorate the tension between predicted and observed small scale structures in the Universe. We argue that coherent forward scattering of neutrinos on fuzzy dark matter particles can significantly alter neutrino oscillation probabilities. These effects could be observable in current and future experiments. We set new limits on fuzzy dark matter interacting with neutrinos using T2K and solar neutrino data, and we estimate the sensitivity of reactor neutrino experiments and of future long-baseline accelerator experiments. These results are based on detailed simulations in GLoBES. We allow the dark matter particle to be either a scalar or a vector boson. In the latter case, we find potentially interesting connections to models addressing various B physics anomalies.