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The virtual element method has been developed over the last decade and applied to problems in elasticity and other areas. The successful application of the method to linear problems leads naturally to the question of its effectiveness in the nonlinear regime. This work is concerned with extensions of the virtual element method to problems of compressible and incompressible nonlinear elasticity. Low-order formulations for problems in two dimensions, with elements being arbitrary polygons, are considered: for these, the ansatz functions are linear along element edges. The various formulations considered are based on minimization of energy, with a novel construction of the stabilization energy. The formulations are investigated through a series of numerical examples, which demonstrate their efficiency, convergence properties, and for the case of nearly incompressible and incompressible materials, locking-free behaviour.