Polynomial Identity Rings A ring R satisfies a polynomial identity if there is a polynomial f in noncommuting variables which vanishes under substitutions from R. For example commutative rings satisfy the polynomial f(x y) = xy yx and exterior algebras satisfy the polynomial f(x y z) = (xy yx)z z(xy yx). Satisfying a polynomial identity is often regarded as a generalization of commutativity. These lecture notes treat polynomial identity rings from both the combinatorial and structural points of view. The former studies the ideal of polynomial identities satisfied by a ring R. The latter studies the properties of rings which satisfy a polynomial identity. The greater part of recent research in polynomial identity rings is about combinatorial questions and the combinatorial part of the lecture notes gives an up to date account of recent research. On the other hand the main structural results have been known for some time and the emphasis there is on a presentation accessible to newcomers to the subject. The intended audience is graduate students in algebra and researchers in algebra combinatorics and invariant theory. A Combinatorial Aspects in PI Rings Vesselin Drensky 1 Basic Properties of PI algebras 2 Quantitative Approach to PI algebras 3 The Amitsur Levitzki Theorem 4 Central Polynomials for Matrices 5 Invariant Theory of Matrices 6 The Nagata Higman Theorem 7 The Shirshov Theorem for Finitely Generated PI algebras 8 Growth of Codimensions of PI algebras B Polynomial Identity Rings Edward Formanek 1 Polynomial Identities 2 The Amitsur Levitzki Theorem 3 Central Polynomials 4 Kaplansky’s Theorem 5 Theorems of Amitsur and Levitzki on Radicals 6 Posner’s Theorem 7 Every PI ring Satisfies a Power of the Standard Identity 8 Azumaya Algebras 9 Artin’s Theorem 10 Chain Conditions 11 Hilbert and Jacobson PI Rings 12 The Ring of Generic Matrices 13 The Generic Division Ring of Two 2 x 2 Generic Matrices 14 The Center of the Generic Division Ring 15 Is the Center of the Generic Division Ring a Rational Function Field.