  Cohomological field theories (CohFTs) were defined in the mid 1990s by
Kontsevich and Manin to capture the formal properties of the virtual
fundamental class in Gromov-Witten theory. A beautiful classification result
for semisimple CohFTs (via the action of the Givental group) was proven by
Teleman in 2012. The Givental-Teleman classification can be used to explicitly
calculate the full CohFT in many interesting cases not approachable by earlier
methods. My goal here is to present an introduction to these ideas together
with a survey of the calculations of the CohFTs obtained from Witten's classes
on the moduli spaces of r-spin curves, Chern characters of the Verlinde bundles
on the moduli of curves, and Gromov-Witten classes of the Hilbert schemes of
points of the plane. The subject is full of basic open questions.
