n this talk, we examine finite element approximations of several mixed formulations of linear elasticity, namely the displacement--pressure, displacement--stress, and displacement--pressure--stress formulations, as well as in flow problems such as viscoelasticity. As in any mixed formulation involving unknowns belonging to different functional spaces, the well-posedness of the problem relies on a global inf-sup condition, which in turn follows from a set of ``little'' inf-sup conditions that must be satisfied by the discrete interpolation spaces associated with the different unknowns. Finite element approximations fulfilling these conditions are often difficult to construct and implement, and in some cases extremely rare, as in the displacement (or velocity)--pressure--stress formulation. An alternative approach consists in modifying the discrete problem by introducing suitable stabilisation terms into the Galerkin formulation, leading to stable approximations for arbitrary choices of interpolation spaces. We discuss the structure of these stabilised formulations and their relevance beyond the context of linear problems. The three-field formulation was originally introduced and fully analysed for the Stokes problem in [1]. The advantages of treating the stress as an independent variable were subsequently highlighted in a series of works beginning with [2], and this approach has since been successfully applied to a wide range of problems involving both linear and nonlinear constitutive laws under the small-strain assumption. Extensions to finite-strain elasticity were reviewed in [3]. This approach has also been used in viscoelastic flows. The objective of this presentation is to highlight the intrinsic difficulties associated with the use of classical inf-sup stable finite element approximations in mixed problems, both in solid and in fluid mechanics, and the flexibility provided by stabilised formulations.