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By a φ -variety V , we mean a supervariety or a ⁎-variety generated by an associative algebra over a field F of characteristic zero. In this case, we consider its sequence of φ -codimensions c n φ (V) and say that V is minimal of polynomial growth n k if c n φ (V) grows like n k , but any proper φ -subvariety grows like n t with t < k. In this paper, we deal with minimal φ -varieties generated by unitary algebras and prove that for k ≤ 2 there is only a finite number of them. We also explicit a list of finite dimensional algebras generating such minimal φ -varieties. For k ≥ 3 , we show that the number of minimal φ -varieties can be infinity and we classify all minimal φ -varieties of polynomial growth n k by giving a recipe for the construction of their T φ -ideals. [ABSTRACT FROM AUTHOR]