Abstract: Algebras with polynomial identities generalize commutative and finite dimensional algebras. This generalization is not only formal. PI-algebras share many structural properties with commutative and finite dimensional algebras. We first recall the notion of a PI-algebra and give some key examples. We then expand the class of PI-algebras to include algebras with certain permutational and rewritable properties. After discussing the Kurosh's problems in ring theory, we present a quantitative version of Shirshov's notion of height which leads to purely combinatorial proofs of the Kurosh's problems for PI-algebras. This method also yields a quantitative proof of Berele's theorem that the Gelfand-Kirillov dimension of the finitely generated PI-algebras is finite. On the other hand, this method works with algebras only having the permutational property, so Kurosh's problems and Berele's theorem have positive solutions for these algebras.