This paper considers the problem of regressing data points on the Grassmann manifold over a scalar-valued variable. The Grassmannian has recently gained considerable attention in the vision community with applications in domain adaptation, face recognition, shape analysis, or the classification of linear dynamical systems. Motivated by the success of these approaches, we introduce a principled formulation for regression tasks on that manifold. We propose an intrinsic geodesic regression model generalizing classical linear least-squares regression. Since geodesics are parametrized by a starting point and a velocity vector, the model enables the synthesis of new observations on the manifold. To exemplify our approach, we demonstrate its applicability on three vision problems where data objects can be represented as points on the Grassmannian: the prediction of traffic speed and crowd counts from dynamical system models of surveillance videos and the modeling of aging trends in human brain structures using an affine-invariant shape representation.