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We develop a theory of levels for irreducible representations of symmetricgroups of degree $n$ analogous to the theory of levels for finite classicalgroups. A key property of level is that the level of a character, provided itis not too big compared to $n$, gives a good lower bound on its degree, and,moreover, every character of low degree is either itself of low level orbecomes so after tensoring with the sign character. Furthermore, if $l_1$ and$l_2$ satisfy a linear upper bound in $n$, then the maximal level ofcomposition factors of the tensor product of representations of levels $l_1$and $l_2$ is $l_1+l_2$. To prove all of this in positive characteristic, wedevelop the notion of rank, which is an analogue of the notion of rank ofcross-characteristic representations of finite classical groups. We show, usingmodular branching rules and degenerate affine Hecke algebras, that the leveland the rank agree, as long as the level is not too large. We exploitSchur-Weyl duality, modular Littlewood-Richardson coefficients and tiltingmodules to prove a modular analogue of the Murnaghan-Littlewood theorem onKronecker products for symmetric groups. As an application, we obtainrepresentation growth results for both ordinary and modular representations ofsymmetric and alternating groups analogous to those for finite groups of Lietype.

Level, rank, and tensor growth of representations of symmetric groups