Rare and/or extreme events in weather and climate often have particularly important implications for human welfare. Despite this, these events are often poorly understood since they are difficult to simulate and are, by definition, rarely observed. In this thesis, we present and apply rare event algorithms to both better characterize and better simulate specific rare events in the ocean and atmosphere systems. First, we use diffusion maps and spectral clustering, a machine learning technique, to better characterize the large and small meanders of the Kuroshio current near Japan. This current has long been considered to be bimodal; however, there have been few data-driven efforts to confirm this bimodality or to characterize the predominant states. By applying the diffusion maps and spectral clustering algorithm in an oceanographic context for the first time, we show that the Kuroshio is indeed bimodal but that the two most common states are characterized by high and low variability rather than by current location (as was previously thought to be the case). We also show that the meanders correlate with the location of a nearby recirculation gyre, thereby providing evidence for a meander transition mechanism that depends on the movement of this gyre. Second, we apply action minimization to the study of tropical cyclone rapid intensification by perturbing the Weather Research and Forecasting model into forming more intense storms than it otherwise would. We show that, compared to ensemble methods commonly used in the study of intensification, this method yields significant computational savings in accessing the tail of the intensification distribution. Action minimization generates maximum likelihood pathways of intensification, thereby allowing us to characterize the preferred intensification mechanisms in the model. We find that action minimization chooses physically realistic intensification mechanisms including low-level heating and the reduction of vertical wind shear. Further, we show that asymmetric heating can cause significantly more intensification than purely symmetric heating and discover a regime of non-linear storm response to asymmetric heating that has not been previously observed.