This course will provide a modern point of view on the wide world of large $N$ conformal field theory (CFT) in diverse spacetime dimensions, placed in the context of the conformal bootstrap and the AdS/CFT Correspondence.\\ Some questions we will tackle: \\ What is a ``Holographic CFT''? \\ -- When does a CFT give rise to an equivalent, emergent simple gravity description? \\ -- What is the conformal bootstrap, and how is it used to constrain and classify the space of consistent large $N$ CFTs and theories of quantum gravity? \\ -- Is string theory the only choice? How can we detect strings and extra dimensions from properties of field theories? \\ -- What precise predictions does quantum gravity seem to make about the space of strongly coupled CFTs? \\ Provisional lecture plan: \\ Lectures I-II: Introducing CFTs and their data, large $N$ conformal bootstrap, AdS/CFT, and the notion of ``Holographic CFT'' \\ Lecture III: Correlation functions and holography \\ Lecture IV: Bootstrapping the space of Holographic CFTs \\ Lecture V: Special focus: Two-dimensional CFTs and AdS$_3$ gravity \\ \\ Course website: https://courses.ipht.fr/?q=en/node/310 \\ \\ Videoconference and in person in Salle Itzykson, IPhT \\ Livestream on youtube.com/IPhT-TV: no subscription required