Resumen: A general spectral theory of a linear operator does not exist, it only exists for particular subclasses of operators. One of the most celebrated theories of this type is the Nagy-Foias¸ spectral theory of Hilbert space contractions, obtained in the 1960ies. It is based on the construction of a functional model, which heavily relies on certain chapters of Complex Analysis, such as the theory of Hardy spaces. Being a contraction on a Hilbert space is characterized by a very simple operator inequality. In a landmark work of 1982, Agler showed how to pass from some other operator inequalities to a functional model of an operator by applying more general reproducing kernel Hilbert spaces instead of Hardy spaces. This work motivated an extensive research, which has become a rapidly growing branch of Operator Theory. I will explain how to construct an explicit functional model for an operator satisfying a rather general operator inequality, and discuss the uniqueness of this model. Some spectral consequences in the spirit of the Nagy-Foias¸ theory will be derived. I will also explain a new connection with the ergodic theory of linear operators.