We report here on the outburst onset and evolution of the new soft gamma-ray repeater SGR 0501+4516. We monitored the new
SGR with XMM- Newton starting on 2008 August 23, 1 day after the source became burst active, and continuing with four more
observations in the following month, with the last one on 2008 September 30. Combining the data with the Swift X-ray telescope
(Swift-XRT) and Suzaku data, we modelled the outburst decay over a 3-month period, and we found that the source flux decreased
exponentially with a time-scale of t(c) = 23.8 d. 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 behaviour of the source during the first month of the outburst decay, 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 second
and fourth 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 similar to 100 keV 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 [(sic) = -1.6(4) x 10(-19) s s(-2)]. Thanks to the phase connection, we were
able to study 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 similar to 80 times lower than that measured during the outburst,
and a rather soft, thermal spectrum.