Offline reinforcement learning (RL) aims to optimize policy from large pre-recorded datasets without interaction with the environment. This setting offers the promise of utilizing diverse and static datasets to obtain policies without costly, risky, active exploration. However, commonly used off-policy deep RL methods perform poorly when facing arbitrary off-policy datasets. In this work, we show that there exists an estimation gap of value-based deep RL algorithms in the offline setting. To eliminate the estimation gap, we propose a novel offline RL algorithm that we term Pessimistic Offline Policy Optimization (POPO), which learns a pessimistic value function. To demonstrate the effectiveness of POPO, we perform experiments on various quality datasets. And we find that POPO performs surprisingly well and scales to tasks with high-dimensional state and action space, comparing or outperforming tested state-of-the-art offline RL algorithms on benchmark tasks.