The framework of Finite Element Systems (FES) was introduced in 2005 (appeared in M3AS 2008) as auxiliary spaces used to define lowest order discrete de Rham sequences on polytopal meshes. Later it was realized that they can be used to analyze existing finite element de Rham sequences (joint work with F. Rapetti and A. Gillette, 2016) in a new and compact way. Finally the framework was generalized to cover other differential operators (such as in elasticity), and smoothness conditions (such as for Stokes) (joint work with K. Hu, 2020, 2023). FES provides an alternative point of view to Ciarlet’s definition of a finite element as a triple with degrees of freedom. It is expressed in the language of category theory (sheaves) and captures more geometric information (trace complexes). I will review this development.