Abstract: The Gromov width of a 2n-dimensional symplectic manifold is the supremum over the numbers \pi R^2 such that the standard 2n-dimensional ball of radius R symplectically embeds in the manifold. Consider the Grassmannian of complex k-planes in C^n, with its symplectic form normalized so that it generates the integral cohomology in degree 2. We show that its Gromov width is equal to one. In fact, we use equivariant techniques to prove that the Gromov width is at least one, and we use holomorphic techniques to prove that the Gromov width is at most one. The lower bound is a special case of more general results: for instance, if a compact manifold with an integral symplectic form admits a Hamiltonian circle action which is semi-free near an isolated minimum for the moment map then its Gromov width is at least one. This is joint work with Sue Tolman.