In the last years, the sequence of codimensions of PI-algebras has been studied by several authors and the classification of unitary algebras, up to equivalence, with at most cubic codimension growth was given by Giambruno, La Mattina and Petrogradsky in 2007. In this paper, we establish a new approach by studying the possibilities of specific proper codimensions of a unitary algebra with growth [Formula: see text] in order to present a complete list of varieties generated by unitary algebras with polynomial growth [Formula: see text]. Also, we classify, up to PI-equivalence, the unitary algebras with growth [Formula: see text] whose leading coefficient of the polynomial describing the codimension sequence achieves the largest and the smallest possible value.