This is a survey of some recently explored features of finite element (FE) models for three fundamental saddle point systems of differential equations, for which an equilibrium divergence constraint occurs. Pairs of FE subspaces in \( H(\mathrm{{div}},\Omega )\times L^2(\Omega )\) are used, the divergence compatibility of them playing a crucial role for stable results. These are the kinds of approximations for flux and pressure fields in classical mixed FE formulations of flows in porous media and for velocity and pressure in recently proposed FE models for Stokes-Brinkman flows. In the latter case, continuity of tangential velocity over interfaces is weakly enforced by a traction multiplier. Approximate stress and displacement fields are also searched in divergence-compatible FE pairs composing stress-mixed elasticity models, tensor symmetry being weakly imposed by the action of a rotation multiplier. Special attention is put on enrichment strategies, by taking approximate H(div) fields with richer resolution inside the elements (or polytopal sub-regions), as compared with coarser normal traces over facets of their boundaries. For stability, the resolution of the remaining variables and multipliers has to be consistently adjusted. This technique is helpful in the design, analysis and efficient implementation of challenging strategies: accuracy enhancement in the presence of distorted elements, hp-adaptivity, coupling of elements of different geometries in the same mesh, and treatment of multiscale circumstances.