Theoretical models of the (d,p) transfer reaction are being exploited for both nuclear astrophysics and spectroscopic studies in nuclear physics. Usually, these direct reaction models use local optical model potentials to describe the nucleon-target and deuteron-target interactions. Within this usual framework, the importance of the deuteron D-state upon low-energy (few MeV per nucleon) reactions is normally associated with spin observables and tensor polarization effects — with very minimal influence on differential cross sections. In contrast, recent work that includes the inherent non-locality of the nucleon optical model potentials  in the Johnson-Tandy adiabatic  three-body model description of the (d,p) transition amplitude, which accounts for deuteron break-up effects, shows sensitivity of the reaction to the large n-p relative momentum content of the deuteron ground-state wave function. The importance of the deuteron D-state component at high n-p relative momenta leads to significant sensitivity of calculated (d,p) cross sections and deduced spectroscopic factors to the choice of deuteron wave function . I will present details of the Johnson-Tandy three-body model of the (d,p) transfer reaction when generalized to include the deuteron D-state and in the presence of nonlocal nucleon-target interactions. Exact calculations within this model will be compared to approximate (leading-order) solutions . The latter approximate solutions can be interpreted in terms of local optical potentials, but evaluated at a shifted value of the energy in the nucleon-target system. This energy shift is shown to increase when including realistic wave functions — with a D-state. I will show the calculated dependence of the D-state effects on the separation energy and orbital angular momentum of the transferred nucleon. The effect on spectroscopic information extracted is also shown for a particular recent case of astrophysical significance .