The structure of the cohomology ring of the Grassmannian has been studied for well over a century, but the analogous questions for the non-maximal isotropic Grassmannians of the symplectic and orthogonal groups are rather unexplored. I will describe a new Giambelli type formula which expresses a general Schubert class on a symplectic Grassmannian as a polynomial in certain special Schubert classes, and some applications. This is joint work with Andrew Kresch and Anders Buch, and grew naturally out of our study of the small quantum cohomology ring of these spaces.