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Let $𝒱$ be a proper variety of associative algebras over a field $F$ of characteristic zero. It is well-known that $𝒱$ can have polynomial or exponential growth and here we present some classification results of varieties of polynomial growth. In particular we classify all subvarieties of the varieties of almost polynomial growth, i.e., the subvarieties of $\text{𝐯𝐚𝐫}(G)$ and $\text{𝐯𝐚𝐫}(U{T}_2)$, where $G$ is the Grassmann algebra and $U{T}_2$ is the algebra of $2\times 2$ upper triangular matrices.

2010 Mathematics Subject Classification: 16R10, 16W55, 16P90. Let V^gr be a variety of associative superalgebras over a field F of characteristic zero. It is well-known that V gr can have polynomial or exponential growth. Here we present some classification results on varieties of polynomial growth. In particular we classify the varieties of at most linear growth and all subvarieties of the varieties of almost polynomial growth. ∗ The author was partially supported by MIUR...