ABSTRACT. A review is given of conditions which characterize when the metric projection onto a proximinal subspace of a normed linear space has a selection which is continuous, (pointwise) Lipschitz continuous, or linear. Intrinsic characterizations of the subspaces in the particular spaces C_{0}(T) or L_{p}(u), 1 ≤ p < ∞, whose metric projections have one of these properties are also given.