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As is the case for all light coloured Standard Model particles, also photons and charged leptons appear as constituents in ultrarelativistic hadron beams, and admit a parton density function (PDF). It has been shown recently that the photon PDF can be given in terms of the structure functions and form factors for electron-proton scattering. The same holds for lepton PDFs. In the present work we set up a calculation of the lepton PDFs at next-to-leading order, using the same data input needed in the photon case. A precise knowledge of the lepton densities allows us to study lepton-initiated processes even at a hadron collider, with all possible combinations of same-charge, opposite-charge, same-flavour, different-flavour leptons and leptons-quarks, most of which cannot be realized in any other foreseeable experiment. The lepton densities in the proton are extremely small, so that their contribution to Standard Model processes is generally shadowed by processes initiated by coloured partons. We will show, however, that there are cases where these processes can be relevant, giving rise to rare Standard Model signatures and to new production channels, that can enlarge the discovery potential of New Physics at the LHC and future high energy colliders with hadrons in the initial state.