Jets, and more recently jet substructure, have been studied since the early days of QCD using observables that measure the shape of radiation produced in collisions. In this talk I will reformulate jet substructure as the study of the near-coincidence (or operator product expansion (OPE)) limit of certain non-local lightray operators. This formulation allows us to build a direct bridge between QCD phenomenology and recently developed techniques in conformal field theory. In particular, I will show that the OPE of lightray operators provides a natural expansion for jet substructure, and predicts rich Lorentzian scaling behavior inside jets at the LHC, which surprisingly has not been studied by the QCD community. Using Open Data from the CMS experiment, I will show that the scaling and shape dependence of matrix elements of multiple lightray operators can be directly measured at the LHC, and exhibit a number of remarkable properties. I will then describe how connections between conformal field theory and jet substructure provide new insights into complicated calculations in jet substructure, and new avenues for better understanding QCD.