This paper formalizes and adapts the well-known concept of Pareto efficiency in the context of the popular robust optimization (RO) methodology for linear optimization problems. We argue that the classical RO paradigm need not produce solutions that possess the associated property of Pareto optimality, and we illustrate via examples how this could lead to inefficiencies and suboptimal 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 limited extra computational cost. We perform numerical experiments drawn from three different application areas (port- folio optimization, inventory management, and project management), which demonstrate that PRO solutions have a significant potential upside compared with solutions obtained using classical RO methods.