We present a comprehensive approach to the formulation and discretization of geometric PDE governing processes relevant in biophysics and materials science. We start with key elements of differential geometry and shape differential calculus which enable us to compute first variations of domain and boundary functionals. We propose geometric gradient flows as a relaxation towards equilibrium and derive the corresponding dynamic equations and their finite element approximation. We apply this framework to mean curvature flow, surface diffusion, and Helfrich flow (Willmore flow with area and volume constraints). We compare the dynamics of biomembranes dictated by either relaxation or an incompressible fluid. We examine the coupling of director fields with flexible surfaces, as occur in models of surfactants and gels. We conclude with large bending deformations with isometry constraint and their application to bilayer actuators.