Supermassive black holes (BHs) have been found in 85 galaxies by dynamical modeling of spatially resolved kinematics. The Hubble Space Telescope revolutionized BH research by advancing the subject from its proof-of-concept phase into quantitative studies of BH demographics. Most influential was the discovery of a tight correlation between BH mass [Formula: see text] and the velocity dispersion σ of the bulge component of the host galaxy. Together with similar correlations with bulge luminosity and mass, this led to the widespread belief that BHs and bulges coevolve by regulating each other's growth. Conclusions based on one set of correlations from [Formula: see text] in brightest cluster ellipticals to [Formula: see text] in the smallest galaxies dominated BH work for more than a decade. New results are now replacing this simple story with a richer and more plausible picture in which BHs correlate differently with different galaxy components. A reasonable aim is to use this progress to refine our understanding of BH-galaxy coevolution. BHs with masses of 105-106M⊙ are found in many bulgeless galaxies. Therefore, classical (elliptical-galaxy-like) bulges are not necessary for BH formation. On the other hand, although they live in galaxy disks, BHs do not correlate with galaxy disks. Also, any [Formula: see text] correlations with the properties of disk-grown pseudobulges and dark matter halos are weak enough to imply no close coevolution. The above and other correlations of host-galaxy parameters with each other and with [Formula: see text] suggest that there are four regimes of BH feedback. (1) Local, secular, episodic, and stochastic feeding of small BHs in largely bulgeless galaxies involves too little energy to result in coevolution. (2) Global feeding in major, wet galaxy mergers rapidly grows giant BHs in short-duration, quasar-like events whose energy feedback does affect galaxy evolution. The resulting hosts are classical bulges and coreless-rotating-disky ellipticals. (3) After these AGN phases and at the highest galaxy masses, maintenance-mode BH feedback into X-ray-emitting gas has the primarily negative effect of helping to keep baryons locked up in hot gas and thereby keeping galaxy formation from going to completion. This happens in giant, core-nonrotating-boxy ellipticals. Their properties, including their tight correlations between [Formula: see text] and core parameters, support the conclusion that core ellipticals form by dissipationless major mergers. They inherit coevolution effects from smaller progenitor galaxies. Also, (4) independent of any feedback physics, in BH growth modes 2 and 3, the averaging that results from successive mergers plays a major role in decreasing the scatter in [Formula: see text] correlations from the large values observed in bulgeless and pseudobulge galaxies to the small values observed in giant elliptical galaxies.