Significant to a latent trait or item response theory analysis of a mental test is the determination of exactly what is being quantified. The following are practical problems to be considered in the formulation of a good theory: (1) deciding whether two tests measure the same trait or traits; (2) analyzing the relative contributions of a pair of traits or abilities to test performance; (3) detecting "functional" changes in items including those caused by security problems, modes of administration changes, and changes in familiarity with the concepts supporting the item in the population being tested; (4) determining the adequacy of an item response function, that is, a specific mathematical formula relating performance to ability; (5) discovering the shape of the item response functions, including multidimensional item response functions; (6) quantifying the magnitude and reliability of violations of the principal assumption of latent trait theory, "local independence"; and (7) modeling item responses (such as omitting or changing answers) that fail to be locally independent. This paper outlines theoretical results bearing on these seven problems. A new theory is presented, with the central problem the representation of traits, abilities or achievements, and their distributions. (PN)