Given a language L, the class of o-minimal L-structures is not elementary, e.g., an ultraproduct of o-minimal structures need not be o-minimal. This fact gives rise to the following notion, introduced by Hans Schoutens: Given a language L, an L-structure is pseudo-o-minimal if it satisfies the common theory of o-minimal L-structures. Of particular importance in pseudo-o-minimal structures are pseudo-finite sets. A definable set in an ordered structure is pseudo-finite if it is closed, bounded, and discrete. Many results from o-minimality translate to pseudo-o-minimality by replacing finite with pseudo-finite. We will review the key role that pseudo-finite sets play in pseudo-o-minimality, as well as other first-order properties of o-minimality such as definable completeness* and local o-minimality**. Finally, we will see how pseudo-finite sets can be used to prove distinctions between generalizations of o-minimality and answer two questions by Schoutens, one of them is whether there is an axiomatization of pseudo-o-minimality by first-order conditions on one-variable formulae only. This also partially answers a conjecture by Antongiulio Fornasiero.