Given dynamics and constraints, the viability kernel gathers points from which there exists an evolution which remains in the constraint set. In this paper, we aim at providing more information: we introduce and study the robustness map which associates each point of the viability kernel with the maximal size of the unexpected disturbance in the state space the system can support now and in the future while always remaining inside the constraint set. We first characterize the hypograph of the robustness map in terms of a viability kernel of an augmented problem. Then the main mathematical result is that the boundary of this hypograph is a viability domain for a particular augmented problem and that the associated regulation map governs increasingly robust evolutions. Given a time horizon, the problem of finding increasingly robust evolutions which reach a given level of robustness is finally solved by the computation of the reaching time of another augmented problem.