We revisit the question of the nature of ultraluminous X-ray sources (ULXs) through a detailed investigation of their spectral shape, using the highest quality X-ray data available in the XMM–Newton public archives (≳10 000 counts in their EPIC spectrum). We confirm that simple spectral models commonly used for the analysis and interpretation of ULXs (power-law continuum and multicolour disc blackbody models) are inadequate in the face of such high-quality data. Instead we find two near ubiquitous features in the spectrum: a soft excess and a rollover in the spectrum at energies above 3 keV. We investigate a range of more physical models to describe these data. Slim discs which include radiation trapping (approximated by a p-free disc model) do not adequately fit the data, and several objects give unphysically high disc temperatures (kTin > 3 keV). Instead, disc plus Comptonized corona models fit the data well, but the derived corona is cool and optically thick (τ∼ 5–30). This is unlike the τ∼ 1 coronae seen in Galactic binaries, ruling out models where ULXs are powered by sub-Eddington accretion on to an intermediate-mass black hole despite many objects having apparently cool disc temperatures. We argue that these observed disc temperatures are not a good indicator of the black hole mass as the powerful, optically thick corona drains energy from the inner disc and obscures it. We estimate the intrinsic (corona-less) disc temperature, and demonstrate that in most cases it lies in the regime of stellar mass black holes. These objects have spectra which range from those similar to the highest mass accretion rate states in Galactic binaries (a single peak at 2–3 keV) to those which clearly have two peaks, one at energies below 1 keV (from the outer, un-Comptonized disc) and one above 3 keV (from the Comptonized, inner disc). However, a few ULXs have a significantly cooler corrected disc temperature; we suggest that these are the most extreme stellar mass black hole accretors, in which a massive wind completely envelopes the inner-disc regions, creating a cool photosphere. We conclude that ULXs provide us with an observational template for the transition between Eddington and super-Eddington accretion flows, with the latter occupying a new ultraluminous accretion state.