In this work, we present a unified framework for finite element exterior calculus in terms of differential forms over n-dimensional manifolds (see e.g. Arnold, 2018) on general polytopal meshes. The De Rham complex of differential forms is discretized on general polytopal meshes. We prove that the sequence of finite-dimensional spaces of differential forms is also a complex – the Discrete De Rham complex – with cohomology groups isomorphic to those of the (continuous) De Rham complex. We also obtain discrete Hemholtz–Hodge decompositions for the operators in the discrete sequence. The main ingredients to achieve this are suitable definitions of the general discrete differential operator and of the corresponding potential. Finally, we provide some examples from fluid mechanics applications (see e.g. Beirão da Veiga et al., 2022). This is a joint work with Daniele A. Di Pietro (Université de Montpellier).