While active galactic nuclei with relativistic jets have long been prime candidates for the origin of extragalactic cosmic rays and neutrinos, the BL Lac object TXS 0506+056 is the first astrophysical source observed to be associated with some confidence ($sim 3sigma$) with a high-energy neutrino, IceCube-170922A, detected by the IceCube Observatory. The source was found to be active in high-energy gamma-rays with Fermi-LAT and in very-high-energy gamma-rays with the MAGIC telescopes. To consistently explain the observed neutrino and multi-wavelength electromagnetic emission of TXS 0506+056, we investigate in detail single-zone models of lepto-hadronic emission, assuming co-spatial acceleration of electrons and protons in the jet, and synchrotron photons from the electrons as targets for photo-hadronic neutrino production. The parameter space concerning the physical conditions of the emission region and particle populations is extensively explored for scenarios where the gamma-rays are dominated by either 1) proton synchrotron emission or 2) synchrotron-self-Compton emission, with a subdominant but non-negligible contribution from photo-hadronic cascades in both cases. We find that the latter can be compatible with the neutrino observations, while the former is strongly disfavoured due to the insufficient neutrino production rate.