With every Coxeter system one can associate a family of algebras considered as deformation of its group algebra. These are so-called Hecke algebras, which are classical objects of study in combinatorics and representation theory. Complex Hecke algebras admit a natural *-structure and a *-representation on Hilbert space. Taking the norm- and SOT-closure in such representation, one obtains Hecke operator algebras, which have recently seen increased attention. \\ \\ In this talk, I will introduce Hecke operator algebras from scratch, focusing on the case of right-angled Coxeter groups. This case is particularly interesting from an operator algebraic perspective, thanks to its description by iterated amalgamated free products. I will survey known results on the structure of Hecke operator algebras, before I describe recent work that clarified the factor decomposition of Hecke von Neumann algebras. Two applications to representation theory will be presented. I will finish with some results on the scope and limits of K-theoretic classification of right-angled Hecke C*-algebras. \\ \\ This is joint work with Adam Skalski.