In the literature, the problem of maximizing the expected discounted reward over all stopping rules has been explicitly solved for a number of reward functions when the underlying process is either a random walk in discrete time or a Levy process in continuous time. All of such reward functions are increasing and logconcave while the corresponding optimal stopping rules have the threshold form (i.e. the solutions are one-sided). In this talk, I will first review the relevant literature and then show that all optimal stopping problems with increasing and logconcave reward functions admit one-sided solutions for general random walks in discrete time and Levy processes in continuous time. (This talk is based on joint work with Yi-Shen Lin.)