  With its large collection area, broad-band energy coverage from optical/NIR
(0.3 to 2.2 micron) to soft/hard X-ray (0.1-600 keV), all-sky monitoring
capability, and on-board follow-up, the proposed Energetic X-ray Imaging Survey
Telescope mission (EXIST, see L. Natalucci contribution at this conference) has
been designed to properly tackle the study of the AGN phenomenon and the role
that SMBH play in the Universe. In particular EXIST will carry out an
unprecedented survey above 10 keV (a factor ~20 increase in hard X-ray
sensitivity compared to current and prior X-ray missions) of SMBH activity, not
just in space but also in time and over a significant expanded energy range;
this strategy will overcome previous selection biases, will break the
"multi-wavelength" identification bottleneck and will dramatically increase the
number of AGN detected above 10 keV that are amenable to detailed follow-up
studies (~50000 AGN are expected). We discuss here on few selected AGN science
topics enabled by the unique combination of EXIST's instruments. In particular
EXIST will enable major progress in understanding: i) when and where SMBH are
active in the Universe (by revealing and measuring heavily obscured accretion
in the local - z<0.5 - Universe), ii) the physics of how SMBH accrete (by
studying the broad-band X-ray spectra and variability properties of an unbiased
and significant sample of AGN), and iii) the link between accretion power and
jet/outflow power (by using observations of blazars). Last but not least
EXIST's ability to find powerful, but very rare blazars, enables it to probe
the appearance of the very first SMBH in the Universe allowing us to set strong
constraints on the models of SMBH formation and early growth in the Universe.
