Recent experimental advances enable high-resolution observations of biological and synthetic active matter across a wide range of length and time scales. A major interdisciplinary challenge is to translate high-dimensional live-imaging and gene-expression data into low-dimensional mathematical models that will allow us to predict and understand the emergent behaviors of complex biophysical systems. In this talk, I will describe our current efforts to develop computational inference frameworks capable of learning interpretable dynamical equations directly from spatio-temporal data provided by our experimental collaborators. After outlining theoretical and computational challenges posed by state-of-the-art sequencing and microscopy data, we will show how symmetry concepts and modern algorithmic approaches can be combined to construct efficient mode representations and robust inference schemes for biophysical model discovery. To illustrate the practical potential, we present example applications ranging from cell migration dynamics and animal locomotion to the collective swarming of active colloids and fish.