AbstractWe study the class of ω‐categorical structures withn‐degenerate algebraic closure for somenε ω, which includes ω‐categorical structures with distributive lattice of algebraically closed subsets (see [4]), and in particular those with degenerate (trivial) algebraic closure. We focus on the models of ω‐categorical universal theories, absolutely ubiquitous structures, and ω‐categorical structures generated by an indiscernible set. The assumption of n‐degeneracy implies total categoricity for the first class, stability for the second, and ω‐sta