  The blazar TXS 0506+056 got into the spotlight of the astrophysical community
in September 2017, when a high-energy neutrino detected by IceCube
(IceCube-170922A) was associated at the 3 $\sigma$ level to a $\gamma$-ray
flare from this source. This multi-messenger photon-neutrino association
remains, as per today, the most significant one ever observed. TXS 0506+056 was
a poorly studied object before the IceCube-170922A event. To better
characterize its broad-band emission, we organized a multi-wavelength campaign
lasting 16 months (November 2017 to February 2019), covering the radio-band
(Mets\"ahovi, OVRO), the optical/UV (ASAS-SN, KVA, REM, Swift/UVOT), the X-rays
(Swift/XRT, NuSTAR), the high-energy $\gamma$ rays (Fermi/LAT) and the
very-high-energy (VHE) $\gamma$ rays (MAGIC). In $\gamma$ rays, the behaviour
of the source was significantly different from the 2017 one: MAGIC observations
show the presence of flaring activity during December 2018, while the source
only shows an excess at the 4$\sigma$ level during the rest of the campaign (74
hours of accumulated exposure); Fermi/LAT observations show several short
(days-to-week timescale) flares, different from the long-term brightening of
2017. No significant flares are detected at lower energies. The radio light
curve shows an increasing flux trend, not seen in other wavelengths. We model
the multi-wavelength spectral energy distributions in a lepto-hadronic
scenario, in which the hadronic emission emerges as Bethe-Heitler and
pion-decay cascade in the X-rays and VHE $\gamma$ rays. According to the model
presented here, the December 2018 $\gamma$-ray flare was connected to a
neutrino emission that was too brief and not bright enough to be detected by
current neutrino instruments.
