A wide variety of natural, technological, and social phenomena are mathematically described in terms of partial differential equations. These are often too complicated to be solved analytically, and this is where numerical methods come into play. Finite elements are a powerful, flexible, and robust class of methods for the numerical approximation of solutions to partial differential equations. In their standard version, they are based on piecewise polynomial functions on a partition of the domain of interest. Continuity requirements are possibly dictated by the regularity of the exact solutions. By breaking the constraints of the classic finite element paradigm, new methods that are specifically tailored to the problem at hand have been developed in order to better reproduce physical properties of the exact solutions, to enhance stability, and to improve accuracy vs. computational cost. Problem-oriented finite elements for the approximation of wave propagation problems will be the focus of this lecture.