A deep reinforcement learning (DRL) agent observes its states through observations, which may contain natural measurement errors or adversarial noises. Since the observations deviate from the true states, they can mislead the agent into making suboptimal actions. Several works have shown this vulnerability via adversarial attacks, but how to improve the robustness of DRL under this setting has not been well studied. We show that naively applying existing techniques on improving robustness for classification tasks, like adversarial training, are ineffective for many RL tasks. We propose the state-adversarial Markov decision process (SA-MDP) to study the fundamental properties of this problem, and develop a theoretically principled policy regularization which can be applied to a large family of DRL algorithms, including deep deterministic policy gradient (DDPG), proximal policy optimization (PPO) and deep Q networks (DQN), for both discrete and continuous action control problems. We significantly improve the robustness of DDPG, PPO and DQN agents under a suite of strong white box adversarial attacks, including two new attacks of our own. Additionally, we find that a robust policy noticeably improves DRL performance in a number of environments.