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Recent progress in an ongoing program of evaluating transverse momentum-dependent (TMD) observables in the proton within Lattice QCD is reviewed. These lattice calculations are based on a definition of TMDs through hadronic matrix elements of quark bilocal operators, including an appropriate gauge connection between the quarks. Results presented include data on the Sivers and Boer-Mulders effects, highlighting the decisive role of the gauge connection in encoding information about the struck quark in a deep inelastic scattering process. First lattice data obtained directly at the physical pion mass are exhibited, and an exploratory outlook on extracting the dependence of the Sivers shift on the quark momentum fraction x is given.