In model theory, given a cardinal $\kappa$, a theory is $\kappa$-categorical (or categorical in cardinality $\kappa$), if it has precisely one isomorphism class of models of cardinality $\kappa$.
The Morley categoricity theorem says that a first-order theory $T$ with countably many symbols is $\kappa$-categorical for one uncountable cardinal $\kappa$ iff $T$ is categorical in any uncountable cardinality.
Study of categoricity lead historically to the development of the stability theory in model theory, see also geometric stability theory.
John Baldwin, Categoricity, Amer. Math. Soc. 2011, pdf
John T. Baldwin, What is a complete theory, talk, pdf
wikipedia: Morley categoricity theorem