  We review conformal field theory on the plane in the conformal bootstrap
approach. We introduce the main ideas of the bootstrap approach to quantum
field theory, and how they apply to two-dimensional theories with local
conformal symmetry. We describe the mathematical structures that appear in such
theories, from the Virasoro algebra and its representations, to BPZ equations
and conformal blocks. Examples include Liouville theory, (generalized) minimal
models, free bosonic theories, the $H_3^+$ model, and the $SU_2$ and
$\widetilde{SL}_2(\mathbb{R})$ WZW models. We also discuss relations between
some of these models, and limits of these models when the central charge and/or
conformal dimensions tend to particular values.
