Abstract: The structure of the cohomology ring of the Grassmannian of linear subspaces of a fixed dimension in a complex vector space has been studied for well over a century, but the analogous questions for the Grassmannians of the symplectic group have been illuminated only recently. I will aim to discuss my work with Anders Buch and Andrew Kresch on the `Giambelli formula' for these spaces, beginning with the early history of such formulas (going back to Jacobi) and their connection to symmetric functions and representation theory. The talk will be accessible to a general audience.