Unitary transformations are the most general input-output maps available in closed quantum systems. Good control protocols have been developed for qubits, but questions remain about the use of optimal control theory to design unitary maps in high-dimensional Hilbert spaces, and about the feasibility of their robust implementation in the laboratory. Here we design and implement unitary maps in a 16-dimensional Hilbert space associated with the 6S(1/2) ground state of Cs-133, achieving fidelities > 0.98 with built-in robustness to static and dynamic perturbations. Our work has relevance for quantum information processing and provides a template for similar advances on other physical platforms.