We propose a model of dynamic trading where a strategic high frequency trader receives an imperfect signal about the future order flow, and exploits his speed advantage to act as a market maker. We determine the provision of liquidity, order cancellations, and impact on low frequency traders. The model predicts that volatility leads high frequency traders to reduce their provision of liquidity. Next, we analyze the problem when the high frequency trader competes with another market maker. Finally, we provide the first formal, model-based analysis of the impact of various policies designed to regulate high frequency trading. Joint with Mehmet Saglam.