The presentation will review recent advances in the area of designing and using polyconvex energy potentials to describe the behaviour of solids in the large strain regime in the presence of several physical phenomena such as thermal or electro-mechanical effects. The need for convexity will be justified from the viewpoint of ensuring the existence of real wave speeds in the material at any state of deformation. The convexity of energy potentials with respect to an extended set of variables describing deformation and other multi-physics effects enables the definition of conjugate stresses, thermal and electro-mechanical variables and the formulation of complementary energy functions via the application of the Legendre transforms. In the static case, this leads to the development of a variety of Hu-Washizu or Hellinger-Reissner mixed variation principles that permit the discretisation of different fields with specifically chosen order of accuracy. For instance, in the case of nearly and fully incompressible materials, discretisations that either meet the LBB condition or are appropriately stabilised can be constructed. In the dynamic case, first order conservation laws can be derived for each of the extended set of variables in the energy function. Moreover, convexity enables the definition of a convex entropy-like functional of the primary conserved variables, leading to a symmetrisation of the conservation laws in terms of conjugate stress-like variables and a robust implementation of a Petrov-Galerkin discretisation process. This can be done in a variety of ways, for instance in the case of thermoelasticity two approaches will be discussed, namely, using the Hamiltonian as “entropy-like” convex function or, alternatively, using the so-called “ballistic free energy” as convex entropy extension. Several examples demonstrating the theoretical developments will be provided.