Extremal sets in the complex plane: Chebotarev, Stahl and Nuttall compacts

The classical Chebotarev problem is about a compact of the minimal logarithmic capacity, connecting a finite number of given points on the complex plane. We discuss the modern applications and generalizations of this problem, such as the «sheet»-structure of the Riemann surface of vector-analytic functions and asymptotics of the Hermite-Padé rational approximants.