A geometrically-exact beam is a nonlinear field-theoretic model for elongated elastic objects. It utilizes moving frames to reduce the number of system’s independent spatial variables, which is a further development of Euler’s approach to the rotational dynamics of rigid bodies. The talk will discuss the dynamics and geometrically-inspired discretization for structure-preserving numerical simulations of free, forced, and constrained beams. This is joint work with Dmitry Zenkov, Donghua Shi, and Zhipeng An.