The measurement of an astrophysical flux of high-energy neutrinos by IceCube is an important
step towards finding the long-sought sources of cosmic rays. Nevertheless, the long exposure
neutrino sky map shows no significant indication of point sources so far. The real-time followup
of neutrino events turned out to be the most successful approach in neutrino point-source searches. It brought, among others, the most compelling evidence for a neutrino point source:
the flaring gamma-ray blazar TXS 0506+056 in coincidence with a single high-energy neutrino
from IceCube (IceCube-170922A). The fast multiwavelength(MWL) follow-up of this alert was
key for establishing this coincidence and constraining the subsequent theoretical modeling for this event. In the long run, accurate and contemporaneous MWL spectral measurements are essential ingredients in investigating the physical processes leading to particle acceleration and emission of radiation. A deeper understanding of those processes allows us to put constraints on the potential neutrino emission. Here we present the light curves and simultaneous spectral energy distributions from November 2017 till February 2021 of MAGIC and MWL monitoring of TXS 0506+056. The more than the two-year-long rise and high state of the radio light curve of TXS 0506+056, which started near the time of the IceCube neutrino detection, seems to have ended, as indicated by a steep decrease in the first half of 2021. We also present the theoretical interpretation of our observations.