The notion of a finite element system is designed to provide an alternative to Ciarlet's definition of a finite element, adapted to the needs of exterior calculus. It allows for cellular decompositions of space (rather than just simplexes or products thereof) and general functions (rather than just polynomials) yet guarantees compatibility with the exterior derivative and existence of commuting interpolation operators. We review basic definitions and properties. As an application, we show how a form of upwinding, compatible with the exterior derivative, can be carried out within this framework. References: S. H. Christiansen, H. Z. Munthe-Kaas, B. Owren. Topics in structure-preserving discretization. Acta Numer. 20 (2011), 1-119. S. H. Christiansen. Upwinding in finite element systems of differential forms. Smale lecture, Proceedings of FoCM 2011, to appear.