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These are the lecture notes for a course on exceptional polynomials taught atthe \textit{AIMS-Volkswagen Stiftung Workshop on Introduction to OrthogonalPolynomials and Applications} that took place in Douala (Cameroon) from October5-12, 2018. They summarize the basic results and construction of exceptionalpoynomials, developed over the past ten years. In addition, some new resultsare presented on the construction of rational solutions to Painlev\'e equationPIV and its higher order generalizations that belong to the$A_{2n}^{(1)}$-Painlev\'e hierarchy. The construction is based on dressingchains of Schr\"odinger operators with potentials that are rational extensionsof the harmonic oscillator. Some of the material presented here(Sturm-Liouville operators, classical orthogonal polynomials, Darboux-Crumtransformations, etc.) are classical and can be found in many textbooks, whilesome results (genus, interlacing and cyclic Maya diagrams) are new andpresented for the first time in this set of lecture notes.

Lectures on exceptional orthogonal polynomials and rational solutions to Painlev\'e equations