States of 137Xe and 135Xe have been investigated via the 136Xe (d,p) and 136Xe (d,t) reactions with 13-MeV incident deuterons and an over-all energy resolution of 45 keV. Q values of 1.637+/-0.020 and -1.723+/-0.020 MeV have been obtained for the respective ground state 136Xe (d,p)137Xe and 136Xe (d,t)135Xe reactions. The angular distribution data have been analyzed using finite-range distorted-wave Born-approximation (DWBA) calculations corrected for nonlocality of the optical potential to extract spectroscopic information. Spin and parity assignments, excitation energies, and spectroscopic factors for most of the observed levels are presented. A 136Xe (d,p) excitation function showed no significant evidence for an anomaly in the (d,p0) cross section near the threshold of the (d,n) reaction to the corresponding isobaric analog state.