The exceptional thermal properties of graphene have attracted enormous interest over the past decade. Consequently, there is an increased push towards utilizing graphene in thermal-management systems such as thermal interface materials and thermoelectrics. While, many recent investigations have demonstrated the ability to dramatically reduce thermal conductivity by molecular functionalization or by the introduction of structural defects, so far, the ability to enhance thermal conductivity of graphene has not been successfully demonstrated in a controllable fashion. For the first time, using robust atomistic simulation techniques such as molecular dynamics (MD) that use spatially resolved heat-current auto-correlation functions and spectral energy density analysis, we show that graphene's thermal conductivity can be suitably tuned to be either higher or lower, by the introduction of structural defects such as vacancies and by proper functionalization of the graphene sheet. Specifically, by varying the periodicity and density of structural defects as well as adsorbed molecules such as C60, we show that the thermal conductivity can be enhanced by up to 10 % and reduced by much larger percentages. In particular, the ability to enhance thermal conductivity is attributed to an increase in optical phonon lifetime, as a result of suppression of out-of-plane phonon-modes and coherent scattering effects, while the reduction in thermal conductivity is primarily attributed to boundary-scattering.