Research into applying finite volume methods to problems in solid mechanics is presented. The investigation has been motivated by the idea that a high-fidelity aeroelastic simulation can be made possible by extending the finite volume methods already implemented in many existing computational fluid dynamics solvers to solid mechanics. A detailed study and survey of applicable methods have been conducted to identify suitable candidates for a three-dimensional time-accurate large-deformation solid mechanics solver. A vertex-centred and a cell-centred solver were initially implemented. Upon comparison, the cell-centred solver appeared to be computationally more efficient despite suffering from a numerical issue. From this analysis, an improved formulation was subsequently developed for the cell-centred solver. In addition, a robust method for numerical gradient evaluation has been developed and incorporated to minimise errors associated with skewed cells or poorly supported cells at domain boundary. The method is based upon the Green-Gauss method, but takes advantage of a secondary numerical gradient to improve overall accuracy and reliability . The improved cell-centred solver has been shown to be accurate, robust and suitable for large deformations in a non-linear validation case against NASTRAN. High computational efficiency has also been demonstrated in OpenMP parallelisation and convergence acceleration via multigrid. The multigrid implementation has been made possible through incorporating a dual-time implicit scheme into the solver.