Policy gradients methods are perhaps the most widely used class of reinforcement learning algorithms. These methods apply to complex, poorly understood, control problems by performing stochastic gradient descent over a parameterized class of polices. Unfortunately, even for simple control problems solvable by classical techniques, policy gradient algorithms face non-convex optimization problems and are widely understood to converge only to local minima. This work identifies structural properties -- shared by finite MDPs and several classic control problems -- which guarantee that policy gradient objective function has no suboptimal local minima despite being non-convex. When these assumptions are relaxed, our work gives conditions under which any local minimum is near-optimal, where the error bound depends on a notion of the expressive capacity of the policy class.