This paper formalizes and adapts the well known concept of Pareto efficiency in the context of the popular robust optimization (RO) methodology. We argue that the classical RO paradigm need not produce solutions that possess the associated property of Pareto optimality, and illustrate via examples how this could lead to inefficiencies and sub-optimal performance in practice. We provide a basic theoretical characterization of Pareto robustly optimal (PRO) solutions, and extend the RO framework by proposing practical methods that verify Pareto optimality, and generate solutions that are PRO. Critically important, our methodology involves solving optimization problems that are of the same complexity as the underlying robust problems, hence the potential improvements from our framework come at essentially no computational cost. We perform numerical experiments drawn from three different application areas (portfolio optimization, inventory management, and project management), which demonstrate that PRO solutions have a significant upside compared with solutions obtained via classical RO methods, at no extra cost or downside.