We study a sample of 61submillimetre galaxies (SMGs) selected from ground-based surveys, with known spectroscopic redshifts and observed with the Herschel Space Observatory as part of the PACS Evolutionary Probe (PEP) and the Herschel Multi-tiered Extragalactic Survey (HerMES) guaranteed time key programmes. Our study makes use of the broad far-infrared and submillimetre wavelength coverage (100-600 μm) only made possible by the combination of observations from the PACS and SPIRE instruments aboard the Herschel Space Observatory. Using a power-law temperature distribution model to derive infrared luminosities and dust temperatures, we measure a dust emissivity spectral index for SMGs of β = 2.0 ± 0.2. Our results unambiguously unveil the diversity of the SMG population. Some SMGs exhibit extreme infrared luminosities of s1013{L&sun;} and relatively warm dust components, while others are fainter (a few times 1012 L&sun;) and are biased towards cold dust temperatures. Although at zs2 classical SMGs (>5 mJy at 850 μm) have large infrared luminosities (s1013 L&sun;), objects only selected on their submm flux densities (without any redshift informations) probe a large range in dust temperatures and infrared luminosities. The extreme infrared luminosities of some SMGs (LIR ≳ 1012.7 L&sun;, 26/61 systems) imply star formation rates (SFRs) of >500 M&sun; yr-1 (assuming a Chabrier IMF and no dominant AGN contribution to the FIR luminosity). Such high SFRs are difficult to reconcile with a secular mode of star formation, and may instead correspond to a merger-driven stage in the evolution of these galaxies. Another observational argument in favour of this scenario is the presence of dust temperatures warmer than that of SMGs of lower luminosities (s40 K as opposed to s25 K), consistent with observations of local ultra-luminous infrared galaxies triggered by major mergers and with results from hydrodynamic simulations of major mergers combined with radiative transfer calculations. Moreover, we find that luminous SMGs are systematically offset from normal star-forming galaxies in the stellar mass-SFR plane, suggesting that they are undergoing starburst events with short duty cycles, compatible with the major merger scenario. On the other hand, a significant fraction of the low infrared luminosity SMGs have cold dust temperatures, are located close to the main sequence of star formation, and therefore might be evolving through a secular mode of star formation. However, the properties of this latter population, especially their dust temperature, should be treated with caution because at these luminosities SMGs are not a representative sample of the entire star-forming galaxy population. Herschel is an ESA space observatory with science instruments provided by European-led Principal Investigator consortia and with important participation from NASA.Tables 1-13 and Appendix A are available in electronic form at http://www.aanda.org