Title: Online Learning under Adversarial Corruptions. Abstract: We study the design of efficient online learning algorithms tolerant to adversarially corrupted rewards. In particular, we study settings where an online algorithm makes a prediction at each time step, and receives a stochastic reward from the environment that can be arbitrarily corrupted with probability $\epsilon \in [0,\frac 1 2)$. Here $\epsilon$ is the noise rate the characterizes the strength of the adversary. As is standard in online learning, we study the design of algorithms with small regret over a period of time steps. However, while the algorithm observes corrupted rewards, we require its regret to be small with respect to the true uncorrupted reward distribution. We build upon recent advances in robust estimation for unsupervised learning problems to design robust online algorithms with near optimal regret in  three different scenarios: stochastic multi-armed bandits, linear contextual bandits, and Markov Decision Processes~(MDPs) with stochastic rewards and transitions. Finally, we provide empirical evidence regarding the robustness of our proposed algorithms on synthetic and real datasets.