  We report here on the outburst onset and evolution of the new Soft Gamma
Repeater SGR 0501+4516. We monitored the new SGR with XMM-Newton starting on
2008 August 23, one day after the source became burst-active, and continuing
with 4 more observations, with the last one on 2008 September 30. Combining the
data with the Swift-XRT and Suzaku data, we modelled the outburst decay over
160 days, and we found that the source flux decreased exponentially with a
timescale of t_c=23.8 days. In the first XMM-Newton observation a large number
of short X-ray bursts were observed, the rate of which decayed drastically in
the following observations. We found large changes in the spectral and timing
behavior of the source during the outburst, with softening emission as the flux
decayed, and the non-thermal soft X-ray spectral component fading faster than
the thermal one. Almost simultaneously to our XMM-Newton observations (on 2008
August 29 and September 2), we observed the source in the hard X-ray range with
INTEGRAL, which clearly detected the source up to ~100keV in the first
pointing, while giving only upper limits during the second pointing,
discovering a variable hard X-ray component fading in less than 10 days after
the bursting activation. We performed a phase-coherent X-ray timing analysis
over about 160 days starting with the burst activation and found evidence of a
strong second derivative period component (\ddot{P} = -1.6(4)x10^{-19}
s/s^{-2}). Thanks to the phase-connection, we were able to study the the
phase-resolved spectral evolution of SGR 0501+4516 in great detail. We also
report on the ROSAT quiescent source data, taken back in 1992 when the source
exhibits a flux ~80 times lower than that measured during the outburst, and a
rather soft, thermal spectrum.
