In December 2004, the soft gamma-ray repeater SGR 1806-20 emitted the most powerful giant flare ever observed. This probably involved a large-scale rearrangement of the magnetosphere leading to observable variations in the properties of its X-ray emission. Here we present the results of the first Suzaku observation of SGR 1806-20, together with almost simultaneous observations with XMM-Newton and INTEGRAL. The source seems to have reached a state characterized by a flux close to the pre-flare level and by a relatively soft spectrum. Despite this, SGR 1806-20 also remained quite active after the giant flare, allowing us to study several short bursts observed by Suzaku in the 1-100 keV range. We discuss the broad-band spectral properties of SGR 1806-20, covering both persistent and bursting emission, in the context of the magnetar model, and consider its recent theoretical developments.