  This article accompanies my lecture at the 2015 AMS summer institute in
algebraic geometry in Salt Lake City. I survey the recent advances in the study
of tautological classes on the moduli spaces of curves. After discussing the
Faber-Zagier relations on the moduli spaces of nonsingular curves and the kappa
rings of the moduli spaces of curves of compact type, I present Pixton's
proposal for a complete calculus of tautological classes on the moduli spaces
of stable curves. Several open questions are discussed. An effort has been made
to condense a great deal of mathematics into as few pages as possible with the
hope that the reader will follow through to the end.
