Liouville field theory: A decade after the revolution

We review recent developments (up to January 2004) of the Liouville field theory and its matrix model dual. This review consists of two parts. In part I, we review the bosonic Liouville theory. After briefly reviewing the necessary background, we discuss the bulk structure constants (the DOZZ formula) and the boundary states (the FZZT brane and the ZZ brane). Various applications are also presented. In part II, we review the supersymmetric extension of the Liouville theory. We first discuss the bulk structure constants and the branes as in the bosonic Liouville theory, and then we present the matrix dual descriptions with some applications. This review also includes some original material such as the derivation of the conjectured dual action for the N = 2 Liouville theory from other known dualities.