The goal of this paper is not to introduce a single algorithm or method, but to make theoretical steps towards fully understanding the training dynamics of generative adversarial networks. In order to substantiate our theoretical analysis, we perform targeted experiments to verify our assumptions, illustrate our claims, and quantify the phenomena. This paper is divided into three sections. The first section introduces the problem at hand. The second section is dedicated to studying and proving rigorously the problems including instability and saturation that arize when training generative adversarial networks. The third section examines a practical and theoretically grounded direction towards solving these problems, while introducing new tools to study them.