Policy gradient (PG) methods are a widely used reinforcement learning methodology in many applications such as video games, autonomous driving, and robotics. In spite of its empirical success, a rigorous understanding of the global convergence of PG methods is lacking in the literature. In this work, we close the gap by viewing PG methods from a nonconvex optimization perspective. In particular, we propose a new variant of PG methods for infinite-horizon problems that uses a random rollout horizon for the Monte-Carlo estimation of the policy gradient. This method then yields an unbiased estimate of the policy gradient with bounded variance, which enables the tools from nonconvex optimization to be applied to establish global convergence. Employing this perspective, we first recover the convergence results with rates to the stationary-point policies in the literature. More interestingly, motivated by advances in nonconvex optimization, we modify the proposed PG method by introducing periodically enlarged stepsizes. The modified algorithm is shown to escape saddle points under mild assumptions on the reward and the policy parameterization. Under a further strict saddle points assumption, this result establishes convergence to essentially locally-optimal policies of the underlying problem, and thus bridges the gap in existing literature on the convergence of PG methods. Results from experiments on the inverted pendulum are then provided to corroborate our theory, namely, by slightly reshaping the reward function to satisfy our assumption, unfavorable saddle points can be avoided and better limit points can be attained. Intriguingly, this empirical finding justifies the benefit of reward-reshaping from a nonconvex optimization perspective.