Numerical methods for nonlinear hyperbolic equations are known for being especially complicated to derive. This comes from the nature of the equations, which may develop nonregular solutions and nonuniqueness of the solution. To guarantee a proper convergence of the discretization, several structures of the system should be ensured: conservativity and entropy inequality are necessary conditions to ensure the right solution is captured. Additionally, the positivity of some physical variables (e.g. density, internal energy), is usually necessary not only for ensuring the physical relevance of the solution, but also for avoiding the computational failure of the code. Among these considerations, some hyperbolic systems are known to include implicit differential constraints, also called involutions. This can be, for example, the conservation of the vorticity for the wave system (linked with the low Mach number accuracy problem), the conservation of the solenoidal character of the magnetic field for the Maxwell system or the Magnetohydrodynamics system, the curl of the deformation tensor for the hyperelastic system in solid mechanics. These involutions are additional constraints with respect to the conservation laws. Numerical schemes that rely on the direct discretization of the conservation laws typically fail to respect these involutions, therefore innovative strategies must be investigated. The aim of this Minisymposium is to review a large spectrum of recent advances in numerical methods for ensuring all the structures that were described in this abstract, and to present examples and applications e.g. to electromagnetics, fluid mechanics and structural mechanics.