ABSTRACT. In this paper we obtain some contributions to the theory of semicontinuity properties of metric projections. This study is along lines analogous to the studies made in [12] and [14]–[16]. In particular, we shall consider four different types of semicontinuity for set-valued mappings: upper and lower semicontinuity — concepts that go back at least to the 1920s (cf. Hahn [4] and the bibliography cited there) — and upper Hausdorff and lower Hausdorff semicontinuity, which were introduced and studied recently in [12].