  We present spatially resolved X-ray spectra taken with the EPIC cameras of
XMM-Newton of a sample of 17 cooling clusters and three non-cooling clusters
for comparison. The deprojected spectra are analyzed with a multi-temperature
model, independent of any a priori assumptions about the physics behind the
cooling and heating mechanisms. All cooling clusters show a central decrement
of the average temperature, most of them of a factor of ~2. At each radius
within the cooling region the gas is non-isothermal. The differential emission
measure distribution peaks near the maximum (ambient) temperature, and steeply
declines towards lower temperatures, proportional to T^3, or alternatively a
cut-off at about a quarter to half of the maximum temperature. In general, we
find a poor correlation between radio flux of the central galaxy and the
temperature decrement of the cooling flow. This is interpreted as evidence that
except for a few cases heating by a central AGN is not the most common cause of
weak cooling flows. We investigate the role of heat conduction by electrons and
find that the theoretically predicted conductivity rates are not high enough to
balance radiation losses. The differential emission measure distribution has
remarkable similarities with the predictions from coronal magnetic loop models.
Also the physical processes involved (radiative cooling, thermal conduction
along the loops, gravity) are similar for clusters loops and coronal loops. If
coronal loop models apply to clusters, we find that a few hundred loops per
scale height should be present. The typical loop sizes deduced from the
observed emission measure distribution are consistent with the characteristic
magnetic field sizes deduced from Faraday rotation measurements.
