We will discuss a classic network revenue management model of Gallego and van Ryzin (1997), which considers a retailer who sells a product (or multiple products) subject to initial inventory constraints over T consecutive periods. Because the optimal policy via dynamic programming is computationally intractable, researchers have proposed various approximate policies for this problem. We are interested in the so-called “re-solving heuristic,” which periodically solves the fluid approximation model. In the quantity-based revenue management setting with discrete types (joint work with P. Bumpensanti), we find that the re-solving heuristic has a worst-case regret of O(T^{1/2}) compared to the optimal policy, whereas a modified re-solving heuristic can achieve uniformly bounded O(1) regret. In the price-based revenue management setting with continuous price sets (joint work with Yining Wang), we show that the re-solving heuristic attains O(1) regret compared to the value of the optimal policy. This improves the O(lnT) regret upper bound established by Jasin (2014). In addition, we prove that there is an Ω(lnT) gap between the value of the optimal policy and that of the fluid model, implying that the fluid model is not an adequate benchmark for constant regret.