We introduce robustness in restless multi-armed bandits (RMABs), a popular model for constrained resource allocation among independent stochastic processes (arms). Nearly all RMAB techniques assume stochastic dynamics are precisely known. However, in many real-world settings, dynamics are estimated with significant uncertainty, e.g., via historical data, which can lead to bad outcomes if ignored. To address this, we develop an algorithm to compute minimax regret--robust policies for RMABs. Our approach uses a double oracle framework (oracles for agent and nature), which is often used for single-process robust planning but requires significant new techniques to accommodate the combinatorial nature of RMABs. Specifically, we design a deep reinforcement learning (RL) algorithm, DDLPO, which tackles the combinatorial challenge by learning an auxiliary "λ-network" in tandem with policy networks per arm, greatly reducing sample complexity, with guarantees on convergence. DDLPO, of general interest, implements our reward-maximizing agent oracle. We then tackle the challenging regret-maximizing nature oracle, a non-stationary RL challenge, by formulating it as a multi-agent RL problem between a policy optimizer and adversarial nature. This formulation is of general interest---we solve it for RMABs by creating a multi-agent extension of DDLPO with a shared critic. We show our approaches work well in three experimental domains.