A fully three-dimensional finite-strain viscoelastic model is developed, characterized by: (i) general anisotropic response, (ii) uncoupled bulk and deviatoric response over any range of deformations, (iii) general relaxation functions, and (iv) recovery of finite elasticity for very fast or very slow processes; in particular, classical models of rubber elasticity (e.g. Mooney-Rivlin). Continuum damage mechanics is employed to develop a simple isotropic damage mechanism which incorporates softening behavior under deformation (Mullins' effect), and leads to progressive degradation of the storage moduli in a cyclic test with increasing amplitude. A numerical integration procedure is presented which preserves exactly the principle of objectivity and by-passes the use of mid-point configurations. Quasi-incompressible response is accounted for within the context of a three-field variational formulation of the Hu-Washizu type.