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

Conformal field theory on the plane