A comprehensive review of the literature on crossover design is needed to highlight its evolution, applications, and methodological advancements across various fields. Given its widespread use in clinical trials and other research domains, understanding this design's challenges, assumptions, and innovations is essential for optimizing its implementation and ensuring accurate, unbiased results. This article extensively reviews the history and statistical inference methods for crossover designs. A primary focus is given to the AB-BA design as it is the most widely used design in literature. Extension from two periods to higher-order designs is discussed, and a general inference procedure for continuous response is studied. Analysis of multivariate and categorical responses is also reviewed in this context. Recent developments, including causal (potential outcomes) formulations of carryover, efficient use of period-specific baselines, generalized estimating equations for repeated measurements within periods, and estimands for incomplete crossover trials, are also discussed. Several open problems in this area are shortlisted.

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