  We present a re-appraisal of the temperatures of Red Supergiants (RSGs) using
their optical and near-infrared spectral energy distributions (SEDs). We have
obtained data of a sample of RSGs in the Magellanic Clouds using VLT+XSHOOTER,
and we fit MARCS model atmospheres to different regions of the spectra,
deriving effective temperatures for each star from (a) the TiO bands, (b)
line-free continuum regions of the spectral energy distributions (SEDs), and
(c) the integrated fluxes. We show that the temperatures derived from fits to
the TiO bands are systematically {\it lower} than the other two methods by
several hundred Kelvin. The TiO fits also dramatically over-predict the flux in
the near-IR, and imply extinctions which are anomalously low compared to
neighbouring stars. In contrast, the SED temperatures provide good fits to the
fluxes at all wavelengths other than the TiO bands, are in agreement with the
temperatures from the flux integration method, and imply extinctions consistent
with nearby stars. After considering a number of ways to reconcile this
discrepancy, we conclude that 3-D effects (i.e.\ granulation) are the most
likely cause, as they affect the temperature structure in the upper layers
where the TiO lines form. The continuum, however, which forms at much deeper
layers, is apparently more robust to such effects. We therefore conclude that
RSG temperatures are much warmer than previously thought. We discuss the
implications of this result for stellar evolution and supernova progenitors,
and provide relations to determine the bolometric luminosities of RSGs from
single-band photometry.
