An appreciable portion of the electricity generated in the United States (U.S.) is provided by nuclear power plants, making the U.S. the largest producer of nuclear power in the world. In order to ensure public safety and effective plant operation, the spent nuclear fuel (SNF) removed from nuclear reactors needs to be stored safely and efficiently. Due to the lack of long-term storage options, the operational period of dry storage structures, originally designed as interim storage solutions, is being extended beyond their intended design life. Among various dry storage structures, this study focuses on vertical concrete dry casks due to their susceptibility to lateral loads, and specifically explores the response of dry casks to seismic and seismically-induced impact events given their alarming performance in past earthquakes, such as the August 23, 2011, earthquake in Virginia. Owing to their design and orientation, dry casks are susceptible to sliding or rocking during seismic events, potentially resulting in collision between adjacent casks or impact damage from tip-over, with concerns regarding structural damage and release of radioactive material. This study explores the seismic behavior of dry casks and their response to tip-over and collision loads by using a probabilistic method that leverages validated finite element models and introduces surrogate modeling of cask behavior in seismic and impact events. Specifically, these surrogate models of key structural responses in the seismic, tip-over and collision events render fragility, sensitivity, and risk analyses for dry casks computationally feasible. By covering a broad parameter space, the models derived afford the opportunity to explore the relative vulnerability of alternative designs to seismic loads, or to consider, for the first time, the influence of temporal parameter variations and aging effects on the risk of cask damage. The probabilistic approach adopted herein enables propagating the uncertainty due to geometric, material and structural parameters as well as the seismic hazard, enhancing seismic risk analysis of this key storage component in the nuclear industry. Along the way to evaluating fragility and risk to dry casks, this thesis uncovers viable surrogate models for estimating the behavior of dry casks under extreme loads, including polynomial response surface models, multivariate adaptive regression splines, regression trees, and support vector machines for regression. Since the duration and response parameters of interest differ significantly in seismic response and impact scenarios, these insights can offer a valuable foundation for future studies that take advantage of surrogate models for structural multi-hazard reliability and risk problems. In the seismic analysis of casks, a new methodology for probabilistic seismic demand modeling is proposed, referred to as a two-layer surrogate modeling strategy. The two-layer approach proposed offers an advance over traditional probabilistic seismic demand modeling methods, by introducing estimation of intermediate predictors with key links to the problem physics, thereby improving the performance of the developed surrogate models for critical response parameters. Although applied in this thesis to dry casks, the approach has potential widespread application to studying the probabilistic seismic response of other rigid-body-type structures, such as bridges with rocking foundations, rigid blocks, statues or laboratory equipment. The models that emerge from this study provide key insights into the important parameters that affect cask response and fragility. Furthermore, they offer an estimate of the risk of exceeding key cask response limits, indicative of potential damage or design concerns. For example, sensitivity analyses on seismic fragility models of the dry casks are conducted to reveal the relative importance of such parameters as the frequency content of the earthquake, friction coefficient, and key geometric parameters on the likelihood of exceeding sliding and rocking limits. Comparative risk estimates across the U.S. integrate the seismic hazard potential with the derived parameterized fragility models to reveal the annual risks of limit state exceedance for alternative designs and cask locations. Probabilistic analysis of impact scenarios explore the role of cask aging and temporal temperature variations in affecting such responses as canister strains or cask accelerations under tip-over and collision events. Given the concern for release of radioactive material to the environment in the event of cask impact, the models provide valuable information for engineers, plant owners and decision makers in the nuclear industry.