  Item response theory (IRT) has become one of the most popular statistical
models for psychometrics, a field of study concerned with the theory and
techniques of psychological measurement. The IRT models are latent factor
models tailored to the analysis, interpretation, and prediction of individuals'
behaviors in answering a set of measurement items that typically involve
categorical response data. Many important questions of measurement are directly
or indirectly answered through the use of IRT models, including scoring
individuals' test performances, validating a test scale, linking two tests,
among others. This paper provides a review of item response theory, including
its statistical framework and psychometric applications. We establish
connections between item response theory and related topics in statistics,
including empirical Bayes, nonparametric methods, matrix completion,
regularized estimation, and sequential analysis. Possible future directions of
IRT are discussed from the perspective of statistical learning.
