In the past decade, neutrino astronomy has established itself with the characterisation of a diffuse astrophysical neutrino flux and the identification of the first candidate sources. Whereas astrophysical neutrinos were expected to provide answers on the origin of cosmic rays, their observation brought new and intriguing questions. In particular, the majority of the diffuse flux remains unexplained. The detection by the KM3NeT experiment of the ultra-high-energy event KM3-230213A marks a new milestone in neutrino astrophysics. With an energy estimated at ~220 PeV, it is the most energetic cosmic neutrino observed to date, opening the question of its astrophysical origin. Blazars, among the most powerful cosmic accelerators, have been proposed as promising sources of both astrophysical neutrinos and ultra-high-energy cosmic rays. Their emission mechanisms naturally favour neutrino spectra peaking at ultra-high energies. Seventeen candidate blazars are identified in the 3° radius error region of KM3-230213A through their multiwavelength signatures. Using archival data and dedicated observations, their properties have been characterised throughout the whole electromagnetic spectrum, from radio to gamma rays. Three sources exhibit flaring behaviour in one of the examined bands, in coincidence with the neutrino arrival time. In this seminar, I will give an overview on the astrophysical followup of KM3-230213A and discuss its potential origin, as well as its implication for neutrino and multimessenger astronomy.