We show that black holes supplied with mass at hyper--Eddington rates drive outflows with mildly sub--relativistic velocities. These are $\sim 0.1 - 0.2c$ for Eddington accretion factors $\dot m_{\rm acc} \sim 10 - 100$, and $\sim 1500\,{\rm km\, s^{-1}}$ for $\dot m_{\rm acc} \sim 10^4$. Winds like this are seen in the X--ray spectra of ultraluminous sources (ULXs), strongly supporting the view that ULXs are stellar--mass compact binaries in hyper--Eddington accretion states. SS433 appears to be an extreme ULX system ($\dot m_{\rm acc} \sim 10^4$) viewed from outside the main X--ray emission cone. For less extreme Eddington factors $\dot m_{\rm acc} \sim 10 - 100$ the photospheric temperatures of the winds are $\sim 100$\, eV, consistent with the picture that the ultraluminous supersoft sources (ULSs) are ULXs seen outside the medium--energy X--ray beam, unifying the ULX/ULS populations and SS433 (actually a ULS but with photospheric emission too soft to detect). For supermassive black holes (SMBHs), feedback from hyper--Eddington accretion is significantly more powerful than the usual near--Eddington (`UFO') case, and if realised in nature would imply $M - \sigma$ masses noticeably smaller than observed. We suggest that the likely warping of the accretion disc in such cases may lead to much of the disc mass being expelled, severely reducing the incidence of such strong feedback. We show that hyper--Eddington feedback from bright ULXs can have major effects on their host galaxies. This is likely to have important consequences for the formation and survival of small galaxies.
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