These are some big problems I know about:
Give a combinatorial description of the Kronecker coefficients.
Find a combinatorial interpretation of the Littlewood-Richardson coefficients for the Jack polynomials, $J_{\mu} J_{\nu} = \sum_\lambda c^\lambda_{\mu\nu}(\alpha) J_\lambda$. It is conjectured (but not proved) that $c^\lambda_{\mu\nu}(\alpha)$ is a polynomial in $\alpha$ with non-negative integer coefficients. (Here, one needs to be a bit careful with which normalization one chooses).
Find a combinatorial description of the multiplicative structure constants for the Schubert polynomials (analogue of the Littlewood-Richardson coefficients in the Schur polynomial case).
The different variants of the shuffle conjecture.
Give a combinatorial formula for the non-homogeneous symmetric Jack polynomials (similar to the Knop-Sahi formula for the ordinary Jack polynomials).