This dissertation extends models of search and exploration to account for strategic interactions typical of important economic environments. Foundational models of search and exploration seldom consider multiple searchers or explorers, mostly treat the options being searched as exogenous, and are often solved with dynamic programming which can obfuscate some properties of the optimal search behaviour. This dissertation extends our understanding of the economic behaviour by incorporating strategic features into these models and studying optimum and equilibrium behaviour. The first chapter studies strategic private exploration games with destructive ties. In a strategic exploration game, players determine the order in which they explore available options, with the objective of maximizing the sum of rewards they discover first. Exploration is private in the sense that players cannot condition the order in which they explore on their competitor's strategy. This chapter describes equilibria in private exploration games where simultaneous discovery eliminates the available reward. Symmetric equilibria in these games are easy to compute and depend on the diversity of expected rewards and the number of competitors. The second chapter outlines optimal allocation when inspection is noisy. Here, a principal receives an unknown reward from allocating to an agent who has private information about the reward. Prior to allocating, the principal may elicit a report from the agent and inspect them at a cost, but must do so without transfers. When the private information is noisy, the mechanism that maximizes the principal's expected return segments signals into two groups, inspects high types, allocating to them only if the inspected return is sufficiently positive, and doesn't inspect low types, compensating them with a small probability of allocation. This relates to a number of applied settings such as employer hiring strategies, public grant mechanisms, and portfolio investment rules. In the final chapter, Weitzman's canonical search problem is mapped into a linear program and Pandora's rule is re-derived. This demonstrates the applicability of the polyhedral approach, and contributes to the adaption of Pandora's rule to a wider class of problems.