Abstract
We begin by reviewing basic notions in Virasoro conformal field theory, including primary and descendant fields, singular vectors, operator product expansions, and the Belavin-Polyakov-Zamolodchikov (BPZ) equation. We then introduce the bosonic free-field realization and show how correlation functions can be reduced to computations with free fields and screening charges. Finally, we discuss a concrete example: the Wess-Zumino-Witten (WZW) model on the sphere, where a free-field (Wakimoto) representation allows us to compute holomorphic correlators as contour integrals. We conclude by discussing current challenges in the subject, including the solution of the Knizhnik-Zamolodchikov (KZ) equations, and by highlighting a few open problems in this direction.