Ultraluminous X-ray sources are extragalactic, off-nucleus, point sources in galaxies, and have X-ray luminosities in excess of 3 × 1039 ergs per second. They are thought to be powered by accretion onto a compact object. Possible explanations include accretion onto neutron stars with strong magnetic fields, onto stellar-mass black holes (of up to 20 solar masses) at or in excess of the classical Eddington limit, or onto intermediate-mass black holes (103–105 solar masses). The lack of sufficient energy resolution in previous analyses has prevented an unambiguous identification of any emission or absorption lines in the X-ray band, thereby precluding a detailed analysis of the accretion flow. Here we report the presence of X-ray emission lines arising from highly ionized iron, oxygen and neon with a cumulative significance in excess of five standard deviations, together with blueshifted (about 0.2 times light velocity) absorption lines of similar significance, in the high-resolution X-ray spectra of the ultraluminous X-ray sources NGC 1313 X-1 and NGC 5408 X-1. The blueshifted absorption lines must occur in a fast-outflowing gas, whereas the emission lines originate in slow-moving gas around the source. We conclude that the compact object in each source is surrounded by powerful winds with an outflow velocity of about 0.2 times that of light, as predicted by models of accreting supermassive black holes and hyper-accreting stellar-mass black holes.