Despite the empirical success of the deep Q network (DQN) reinforcement learning algorithm and its variants, DQN is still not well understood and it does not guarantee convergence. In this work, we show that DQN can indeed diverge and cease to operate in realistic settings. Although there exist gradient-based convergent methods, we show that they actually have inherent problems in learning dynamics which cause them to fail even for simple tasks. To overcome these problems, we propose a convergent DQN algorithm (C-DQN) that is guaranteed to converge and can work with large discount factors (0.9998). It learns robustly in difficult settings and can learn several difficult games in the Atari 2600 benchmark that DQN fails to solve.