Some hyperbolic systems are known for ensuring implicitly a divergence or a curl constraint. For example, the wave system ensures the preservation of the vorticity, or the Maxwell system and the Magnetohydrodynamics system respectively ensure a preservation of the divergence of the magnetic field, and an initially divergence free magnetic field. These constraints are usually complicated to preserve at the discrete level, and it is usual to resort to so-called staggered schemes, e.g. MAC scheme for ensuring the incompressibiity constraint, or the Yee scheme for electromagnetism, and their higher order variants based on Nédélec or Raviart-Thomas finite elements. This talk will start from a remark that was done years ago in the context of low Mach number problems (see Rieper & Bader, 2009 or Guillard, 2009), and based on these results, we will explain in which conditions, and in which sense divergence and curl constraints can be preserved with finite volume schemes on triangular meshes. Then, based on ideas developed within the "finite element exterior calculus" (Arnold, 2018), and on the distributional de-Rham complexes introduced by Licht, 2017, I will show how to extend the low order case on triangular meshes to the high order case, and to the quadrangular meshes case. Results of this talk were published in Jung & Perrier, 2024, Perrier, 2024 and Perrier, 2025.