Tautological algebras of moduli spaces of curves

These are the lecture notes for my course at the 2011 Park City Mathematical Institute on moduli spaces of Riemann surfaces. The two lectures here correspond roughly to the first and second half of the course. The subject of the first lecture is the tautological ring R∗(Mg) of Mg. I recall Mumford’s definition of the tautological classes and some of his results from [48]. Then I discuss my conjecture on R∗(Mg) from [10] and the results obtained on it. Finally, I survey some recent developments indicating that the relations that suffice to prove the conjecture for g ≤ 23 may not suffice for larger g. The second lecture concerns mainly the tautological ring ofMg,n. Some natural spaces in between Mg,n and Mg,n are discussed as well. I close with some recent results regarding non-tautological cohomology classes and the cohomology of Mg,n in low genus.