We give a brief overview of recent progress in lattice QCD simulations that is enabling precision studies of the three-dimensional structure of the nucleon. We present results on nucleon charges and second Mellin moments of parton distribution functions and generalized parton distributions, highlighting results obtained at the continuum limit using only gauge ensembles simulated with physical quark masses. The tensor charge and transversity moments are determined with controlled systematics and used to extract nucleon transverse density distributions. We also discuss progress towards the direct evaluation of generalized parton distributions in lattice QCD and the impact they are having on phenomenology.