In continuum mechanics, particularly at solid mechanics analysis of deformable bodies, metals and their alloys have the largest share, whose constitutive stress and strain equations have been worked out to smallest detail. However, in the last half of 20th century, hyperelastic materials have an increasing presence in practice due to the development and thus increase of knowledge about their behaviours. Content of this thesis is focused on nonlinear solid mechanics respectively, on concept of hyperelastic material and its characteristics, together with resolution procedures. Initially, basic terms related to motion and stresses of bodies with great deformations and volume change are introduced. Important kinematic quantities from the finite strain theory as well as their time change intensities are described. Stress and equlibrium equations have been derived, with description and derivation of principle of virtual work in material and spatial configuration. Alternative stress representations are shown (first and second Piola-Kirchhoff stress) and their conjugate deformation quantities which are used in formulation of principle of virtual work. The notion of hyperelastic material is introduced with expressions that define nonlinear behaviour and large deformations. Some models of hyperelastic materials are also briefly described and their corresponding stored strain energy functions, from which the expressions for coresponding stresses and elasticity tensors follow. Since it is a matter of nonlinear material behavior, it is necessary to apply linearization of constitutive equations in order to implement Newton-Raphson iterative procedure. For this purpose, virtual work equations in material and spatial configuration are linearized. Thereby, every component of linearized virtual work equation represents a contribution to the fundamental equation of finite element formulation: K u=F. The nonlinear finite element method formulation is explained, with discretization of essential kinematic quantities and equilibrium equations with the aim of establishing tangent stiffness matrix K and vectors of equivalent nodal forces. At the very end, the software FLagSHyP was selected for numerical application, which contains a library of necessary functions used to solve nonlinear static problems. Brief instructions regarding the use of software are given and certain examples are solved in order to show and explain nonlinear dependence between stress and strain. Solved examples are described by the same hyperelastic material model with the equal properties with the goal to present behavior of a certain model in different load cases. In conclusion, influence of the properties of hyperelastic material on its stiffness is shown.