O-minimal structures are collections of subsets in ${\mathbb{R}}^n$ that have certain finiteness properties that make them similar to semialgebraic sets. I will start by recalling the definition of the latter and discussing the Tarski--Seidenberg theorem claiming that a linear projection of a semialgebraic set is again semialgebraic. Then I will try to give an overview of how one can generalize this and obtain the notion of an o-minimal structure. I expect that the talk will mostly be informal, since I am not a specialist in o-minimal structures and got interested in them because of their recent applications to Hodge theory.