As predictive models are increasingly being deployed in high-stakes decision making (e.g., loan approvals), there has been growing interest in post-hoc techniques which provide recourse to affected individuals. These techniques generate recourses under the assumption that the underlying predictive model does not change. However, in practice, models are often regularly updated for a variety of reasons (e.g., dataset shifts), thereby rendering previously prescribed recourses ineffective. To address this problem, we propose a novel framework, Robust Algorithmic Recourse (ROAR), that leverages adversarial training for finding recourses that are robust to model shifts. To the best of our knowledge, this work proposes the first ever solution to this critical problem. We also carry out detailed theoretical analysis which underscores the importance of constructing recourses that are robust to model shifts: 1) we derive a lower bound on the probability of invalidation of recourses generated by existing approaches which are not robust to model shifts. 2) we prove that the additional cost incurred due to the robust recourses output by our framework is bounded. Experimental evaluation demonstrates the efficacy of the proposed framework and supports our theoretical findings.