  We review the recent developments in the study of conformal field theories in
generic space time dimensions using the methods of the conformal bootstrap, in
its analytic aspect. These techniques are based solely on symmetries, in
particular in the analytic structure and in the associativity of the operator
product expansion. We focus on two applications of the analytic conformal
bootstrap: the study of the $\epsilon$ expansion of the Wilson Fisher model via
the introduction of a dispersion relation and the large $N$ expansion of
maximally supersymmetric Super Yang Mills theory in four dimensions.
