Superintegrability with higher order Painlevé transcendent potentials

On the last decade, there has been an increasing interest on the superintegrability in classical and quantum Mechanics. In this talk, I will start by briefly reviewing some basic notions of superintegrability and then present some recent results on the classification of superintegrable systems adnitting higher-order potentials. The main emphasis will be on the emergence of the Painlevé transcendent potentials in the quantum case.