I will describe an ongoing project to study slices to Schubert varieties in the affine Grassmannian. These are Poisson varieties, and we will be mainly interested in quantizing them. The resulting algebras, called truncated shifted Yangians, have a beautiful representation theory. We will discuss this and also mention some connections to Nakajima quiver varieties which were recently discovered by Braverman-Finkelberg-Nakajima and Webster. In the pre-talk I'll define the affine Grassmannian and discuss its role in geometric representation theory.