Abstract <p> The concept of a pseudo-countably-categorical formula is introduced, and the syntactic and semantic possibilities of such formulas are investigated. Approximations of theories using such formulas, including pseudo-countably-categorical ordered theories, theories of Abelian groups and modules, and Ehrenfeucht theories, are studied. A description of canonical representatives of pseudo-countably-categorical ordered theories is given. The preservation and violation of pseudo-countable categoricity under expansions and restrictions of theories are considered. Countable spectra of pseudo-countably-categorical theories are described. </p>