Title: Two-Timescale Networks for Nonlinear Value Function Approximation. Abstract: A key component for many reinforcement learning agents is to learn a value function, either for policy evaluation or control. Many of the algorithms for learning values, however, are designed for linear function approximation---with a fixed basis or fixed representation. Though there have been a few sound extensions to nonlinear function approximation, such as nonlinear gradient temporal difference learning, these methods have largely not been adopted, eschewed in favour of simpler but not sound methods like temporal difference learning and Q-learning. In this work, we provide a two-timescale network (TTN) architecture that enables linear methods to be used to learn values, with a nonlinear representation learned at a slower timescale. The approach facilitates the use of algorithms developed for the linear setting, such as data-efficient least-squares methods, eligibility traces and the myriad of recently developed linear policy evaluation algorithms, to provide nonlinear value estimates. We prove convergence for TTNs, with particular care given to ensure convergence of the fast linear component under potentially dependent features provided by the learned representation. We empirically demonstrate the benefits of TTNs, compared to other nonlinear value function approximation algorithms, both for policy evaluation and control.    