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abstract: 'The potentially significant role of the surface of an elastic body in the overall response of the continuum can be described using the mature theory of surface elasticity. The objective of this contribution is to detail the finite element approximation of the underlying governing equations (both in the volume and on its surface) and their solution using the open-source finite element library [deal.II]{}. The fully-nonlinear (geometric and material) setting is considered. The nonlinear problem is solved using a Newton–Raphson procedure wherein the tangent contributions from the volume and surface are computed exactly. The finite element formulation is implemented within the total Lagrangian framework and a Bubnov–Galerkin spatial discretization of the volume and the surface employed. The surface is assumed material. A map between the degrees of freedom on the surface and on the boundary of the volume is used to allocate the contribution from the surface to the global system matrix and residual vector. The [deal.II]{} library greatly facilitates the computation of the various surface operators, allowing the numerical implementation to closely match the theory developed in a companion paper. Key features of the theory and the numerical implementation are elucidated using a series of benchmark example problems. 