Summary: This paper starts with a general introduction into measurement of hypothetical constructs typical of the social and behavioral sciences. After the stages ranging from theory through operationalization and item domain to preliminary test or questionnaire have been treated, the general assumptions of item response theory are discussed. The family of parametric item response models for dichotomous items (e.g., correct/incorrect scores) is introduced and it is explained how parameters for respondents and items are estimated from the scores collected from a sample of respondents who took the test or questionnaire. Next, the family of nonparametric item response models is explained, followed by the three classes of item response models for polytomous item scores (e.g., rating scale scores). Then, it is discussed to what degree the mean item score (the p-value for dichotomous items) and the unweighted sum of item scores for persons (the total test score) are useful for measuring items and persons in the context of item response theory. The concepts of invariant item ordering for items, and monotone likelihood ratio, stochastic ordering, and ordering of the expected latent trait for persons, are relevant here. So far, the paper has concentrated on measurement of properties of persons and items, based on item response models. Such measurements make sense only when the item response model fits the data. Methods for fitting models to data are briefly discussed for parametric and nonparametric models, but also two recent hybrid methods are mentioned. Finally, the main applications of item response models are discussed, which include equating and item banking, computerized and adaptive testing, research into differential item functioning, person fit research, and cognitive modeling.