We provide a brief but self-contained review of conformal field theory on the
Riemann sphere. We first introduce general axioms such as local conformal
invariance, and derive Ward identities and BPZ equations. We then define
minimal models and Liouville theory by specific axioms on their spectrums and
degenerate fields. We solve these theories by computing three- and four-point
functions, and discuss their existence and uniqueness.