This work focuses on the virtual element method (VEM) and its application to hyperelastic contact problems involving large deformations. The focus is placed on both the mathematical formulation of VEM and its effectiveness in contact phenomena in two- and three-dimensional problems. Firstly, a stabilization-free virtual element formulation is developed. By exploiting the high-order projection operator in VEM, the proposed formulation achieves consistency and stability without relying on any stabilization, leading to a clear mathematical structure and improved robustness for nonlinear problems. The formulation is applicable to general polygonal and polyhedral meshes, which provides significant flexibility in handling complex geometries and evolving contact interfaces. Secondly, the developed VEM framework is applied to classical large-deformation contact problems in hyperelasticity using a penalty-based contact algorithm. Numerical examples demonstrate that the method can accurately capture contact pressures, deformation patterns, and stress distributions, while maintaining good convergence properties even on highly irregular meshes. Finally, the virtual element method is extended to contact problems based on the third medium contact theory. Both two-dimensional and three-dimensional contact models are constructed within the VEM framework. The introduction of a third medium avoids the introduction of gap constraints and simplifying the numerical implementation. The flexibility of VEM with respect to mesh topology makes it particularly suitable for this class of contact formulations.