A general framework of latent trait item response models for continuous responses is given. In contrast to classical test theory (CTT) models, which traditionally distinguish between true scores and error scores, the responses are clearly linked to latent traits. It is shown that CTT models can be derived as special cases, but the model class is much wider. It provides, in particular, appropriate modeling of responses that are restricted in some way, for example, if responses are positive or are restricted to an interval. Restrictions of this sort are easily incorporated in the modeling framework. Restriction to an interval is typically ignored in common models yielding inappropriate models, for example, when modeling Likert-type data. The model also extends common response time models, which can be treated as special cases. The properties of the model class are derived and the role of the total score is investigated, which leads to a modified total score. Several applications illustrate the use of the model including an example, in which covariates that may modify the response are taken into account.