The Kronecker coefficients of the Symmetric group Sn are the multiplicities of an irreducible Sn representation in the tensor product of two other irreducibles. They were introduced in 1938 by Murnaghan and generalize the beloved Littlewood-Richardson coefficients of the General Linear group. 86 years later still very little is known about them and they are subject of some of the major open problems in Algebraic Combinatorics. They also appear within Geometric Complexity theory, in the quest for multiplicity obstructions to show computational lower bounds for VP versus VNP. In this talk, we will give an overview of the problems and discuss some recent developments on positivity, asymptotics and computational complexity of the Kronecker coefficients and the reduced Kronecker coefficients. We will feature results from joint works with Christian Ikenmeyer, Igor Pak and Damir Yeliussizov.