<p>This paper formulates, implements, and demonstrates a method for the Topology Optimization (TO) of systems undergoing hydrodynamic responses, such as the large dynamic deformations encountered in impact and shock propagation problems. Specifically, it addresses TO for hydrodynamics modeled with Lagrangian material motion and its numerical implementation using the Finite Element Method. It discusses and addresses several of the challenges of gradient based optimization for non-linear dynamic problems in continuum mechanics: such as the objective formulation, adjoint sensitivity equations, and checkpoint schemes. The method is implemented as a parallel algorithm in the Fierro material simulation and optimization codebase. Two Topology Optimization problems are demonstrated: the minimization of kinetic energy to create a mass limited blast shield for shockwave propagation, and maximizing internal energy in a Taylor Anvil impact problem to create a tougher mass limited design.</p>