In this talk, we study the problem of minimizing the distance from a data point to a subvariety of the Grassmannian from the perspective of metric algebraic geometry. An important invariant in this line of work is the number of critical points of this optimization problem for general data. When the data is a general point in the ambient space, this number is called the Euclidean distance (ED) degree of the variety. However, we focus mainly on the case when the data is a general point in the Grassmannian, and call the number of critical points the Grassmann distance (GD) degree. As we will see, the GD degree is more subtle than the ED degree. We give formulas for the ED and GD degrees of prominent subvarieties of the Grassmannian, including Schubert and matroid varieties.