Aims: The Herschel-ATLAS survey (H-ATLAS) will be the largest area survey to be undertaken by the Herschel Space Observatory. It will cover 550 sq. deg. of extragalactic sky at wavelengths of 100, 160, 250, 350 and 500 μm when completed, reaching flux limits (5σ) from 32 to 145 mJy. We here present galaxy number counts obtained for SPIRE observations of the first ~14 sq. deg. observed at 250, 350 and 500 μm. Methods: Number counts are a fundamental tool in constraining models of galaxy evolution. We use source catalogs extracted from the H-ATLAS maps as the basis for such an analysis. Correction factors for completeness and flux boosting are derived by applying our extraction method to model catalogs and then applied to the raw observational counts. Results: We find a steep rise in the number counts at flux levels of 100-200 mJy in all three SPIRE bands, consistent with results from BLAST. The counts are compared to a range of galaxy evolution models. None of the current models is an ideal fit to the data but all ascribe the steep rise to a population of luminous, rapidly evolving dusty galaxies at moderate to high redshift. Herschel is an ESA space observatory with science instruments provided by European-led Principal Investigator consortia and with important participation from NASA.