Mixed Finite Elements (FE) constitute an elegant remedy for the approximation of constrained boundary value problems, where the capability of the classical FE method is limited. This thesis comprises in a first step the mathematical analysis and numerical investigation of different mixed FE approaches in the case of linear elasticity. In a second step novel strategies for the extension of the considered formulations to the nonlinear hyperelastic framework are discussed. Within the main objective of reliable and efficient FE based approximations including large deformations, a focus of the proposed work is set on the construction of elements based on the Hellinger-Reissner variational framework. This family of elements is characterized by a direct discretization of the stresses as well as the displacements and a challenging extension to the large strain regime. The investigation of the efficiency and reliability of the proposed FE schemes is emphasized by a comparison to well established formulations using nontrivial numerical benchmarks. Additionally to the common constraint situations of incompressibility and thin-walled structures, the important case of inextensibility is regarded. This work results in a couple of novel FE discretizations, which are characterized by a notable gain in efficiency and robustness. In addition, further insights considering the reliability and stability of mixed Finite Elements in the hyperelastic framework are gained.