A binomial regression functionp(x, θ) models the probability ofrjsuccesses innjtrials as a function of the values of an observed covariatexjand/or a latent variableθj(j= 1, …,J). This article explores the use of monotone regression splines to definep, and applies them to the representation of test items as functions of examinee ability. Some illustrative data suggest that the flexibility of monotone splines permits the detection of item characteristics not observable using logistic-based or log-linear approaches. A simulation study indicates that estimates of both item-characteristic curves and ability are reasonably precise for numbers of items and examinees typical of large university lectures. Given a set of such binomial regression functions, it can be useful to study the principal components of functional variation. The extension of multivariate principal-components analysis to permit the analysis of many item-characteristic curves is described.