  Over fourty years ago, the physicist Polyakov proposed a bold framework for
string theory, in which the problem was reduced to the study of certain "random
surfaces". He further made the tantalising suggestion that this theory could be
explicitly solved. Recent breakthroughs from the last fifteen years have not
only given a concrete mathematical basis for this theory but also verified some
of its most striking predictions, as well as Polyakov's original vision. This
theory, now known in the mathematics literature either as Liouville quantum
gravity or Liouville conformal field theory, is based on a remarkable
combination of ideas coming from different fields, above all probability and
geometry. This book is intended to be an introduction to these developments
assuming as few prerequisites as possible.
