I show that Eddington accretion episodes in active galactic nuclei (AGN) are likely to produce winds with velocities v∼ 0.1c and ionization parameters up to ξ∼ 104 (cgs), implying the presence of resonance lines of helium- and hydrogen-like iron. These properties are direct consequences of momentum and mass conservation, respectively, and agree with recent X-ray observations of fast outflows from AGN. Because the wind is significantly subluminal, it can persist long after the AGN is observed to have become sub-Eddington. The wind creates a strong cooling shock as it interacts with the interstellar medium of the host galaxy, and this cooling region may be observable in an inverse Compton continuum and lower excitation emission lines associated with lower velocities. The shell of matter swept up by the (‘momentum-driven’) shocked wind must propagate beyond the black hole's sphere of influence on a time-scale of ≲3 × 105 yr. Outside this radius, the shell stalls unless the black hole mass has reached the value Mσ implied by the M–σ relation. If the wind shock did not cool, as suggested here, the resulting (‘energy-driven’) outflow would imply a far smaller supermassive black hole mass than actually observed. In galaxies with large bulges the black hole may grow somewhat beyond this value, suggesting that the observed M–σ relation may curve upwards at large M. Minor accretion events with small gas fractions can produce galaxy-wide outflows with velocities significantly exceeding σ, including fossil outflows in galaxies where there is little current AGN activity. Some rare cases may reveal the energy-driven outflows which sweep gas out of the galaxy and establish the black hole–bulge mass relation. However, these require the quasar to be at the Eddington luminosity.