  Kostka, Littlewood-Richardson, Kronecker, and plethysm coefficients are
fundamental quantities in algebraic combinatorics, yet many natural questions
about them stay unanswered for more than 80 years. Kronecker and plethysm
coefficients lack ``nice formulas'', a notion that can be formalized using
computational complexity theory. Beyond formulas and combinatorial
interpretations, we can attempt to understand their asymptotic behavior in
various regimes, and inequalities they could satisfy. Understanding these
quantities has applications beyond combinatorics. On the one hand, the
asymptotics of structure constants is closely related to understanding the
[limit] behavior of vertex and tiling models in statistical mechanics. More
recently, these structure constants have been involved in establishing
computational complexity lower bounds and separation of complexity classes like
VP vs VNP, the algebraic analogs of P vs NP in arithmetic complexity theory.
Here we discuss the outstanding problems related to asymptotics, positivity,
and complexity of structure constants focusing mostly on the Kronecker
coefficients of the symmetric group and, less so, on the plethysm coefficients.
  This expository paper is based on the talk presented at the Open Problems in
Algebraic Combinatorics coneference in May 2022.
