Despite their starring role in enabling the miracle of life, micro- and nanoscale biological systems have historically defied the suite of mathematical techniques that has proven "unreasonably effective" at modeling their macroscopic counterparts. The computational challenges that characterize such systems include: 1) a large number of interacting degrees of freedom, rendering many traditional models intractable; 2) sensitive, deformable components that exhibit compositional disorder and nonlinear responses; and 3) active agents capable of self-propulsion, creating systems that are inherently out of equilibrium. Our mission is to illuminate the physical principles governing complex biological and bioinspired systems possessing one or more of these characteristics by synthesizing applied analysis, continuum mechanics, differential geometry, and numerical simulation.