In this talk we discuss developments in a first-order system least squares (FOSLS) approach for the numerical approximation of the solution of the equations of geometrically-nonlinear elasticity. We follow a Newton-FOSLS algorithm where each linear step is solved as a two-stage, first-order system under a least squares finite element discretization. With appropriate regularity we show H¹ equivalence of the quadratic part of the FOSLS functional norm in the case of pure displacement boundary conditions. Results hold for deformations near the reference configuration, a set we show to be largely coincident with the set of deformations allowed by the physical model. In this regime the discrete systems that result from using standard finite element subspaces of H¹ can be solved with optimal complexity. Numerical results are given for both pure displacement and mixed boundary conditions and confirm optimal multigrid performance and finite element approximation properties.