This dissertation introduces new models for the locomotion and control of mechanical and hydrodynamic systems that exhibit symmetry, constraints, and control. I introduce the class of unbalanced Chaplygin control systems and analyze the dynamics and control of two new examples in this class --- the Chaplygin beanie and the Chaplygin pendulum. I then prove for the former, and numerically verify for the latter, that a single-input control strategy is able to generate locomotion and asymptotically control both heading and speed. The resulting underactuated control strategy provides a basis for single-input navigation of planar robots subject to a nonholonomic no-slip constraint. Next, I introduce a new model for the locomotion of articulated rigid bodies in ideal fluids and demonstrate that even in the absence of added-mass effects arising from an asymmetric shape, these swimmers may nevertheless locomote provided symmetry of the fluid boundary is broken. I also show that an underlying geometric phase may govern their motion. I then introduce a new technique for devising reduced-order models of the interaction of infinitesimal, stationary rigid bodies immersed in inviscid incompressible fluids with point vorticity. The rigid bodies considered in this dissertation impose four types of constraints on the fluid flow --- velocity, direction (or tangency), distance, and position constraints. It is found that energy is generally conserved but that linear and angular impulse of the fluid are not, and that the constraints may dramatically alter the system dynamics. I also show that this technique may be used to generate approximate models for the interaction of moving rigid bodies and vortical fluids. Finally, motivated by both the relevance of principal bundles to locomotion and by the mismatch between the data needed to prove important theorems vs. the data available in a typical application, I elucidate the relationship between three definitions of a principal bundle appearing in the literature, introduce a new definition, and demonstrate mutual equivalence of all four.
