The Casorati curvature of a submanifold Mn of a Riemannian manifold Mn+m is known to be the normalized square of the length of the second fundamental form, C = 1 n h 2, i.e., in particular, for hypersurfaces, C = 1 n (k2 1 +· · ·+k2n ), whereby k1, . . . , kn are the principal normal curvatures of these hypersurfaces. In this paper we in addition define the Casorati curvature of a submanifold Mn in a Riemannian manifold Mn+m at any point p of Mn in any tangent direction u of Mn. The principal extrinsic (Casorati) directions of a submanifold at a point are defined as an extension of the principal directions of a hypersurface Mn at a point in Mn+1. A geometrical interpretation of the Casorati curvature of Mn in Mn+m at p in the direction u is given. A characterization of normally flat submanifolds in Euclidean spaces is given in terms of a relation between the Casorati curvatures and the normal curvatures of these submanifolds.