The cohomology ring of the usual Grassmannian has been studied for well over a century, but the corresponding Schubert
calculus questions for the Grassmannians of the symplectic and orthogonal groups are much less explored. I will talk about a
Giambelli type formula for arbitrary symplectic Grassmannians and some first applications. If time permits, I will also give some idea about the proof. Most of this is joint work with Anders Buch and Andrew Kresch.