Abstract This paper is devoted to an analysis of the concept of a psychometric function, from an axiomatic viewpoint. The purpose of our work is to provide a convenient language within which most specific models can be discussed. The basic notion is that of a family { p a } of continuous, strictly increasing psychometric functions x ↦ p a ( x ), where the index a , representing the background or the standard, takes values in a metric space, and the stimulus value x varies in an open real interval. Important general properties of such families (symmetry, parallelism, …) are investigated, and axiom systems for representations such as p a ( x ) = H [ x , μ ( a )], or more specifically p a ( x ) = F [ u ( x ) − g ( a )], are discussed in detail. This last equation generalizes the classical Fechnerian representation as well as the “univariance” model of visual psychophysics. Various particular models are given as illustrative examples of the general theory.