A Riemannian manifold \((M,g)\) equipped with a unit vector field \(\xi\) is of quasi-constant sectional curvatures (a QC-manifold) if at each point \(p\), the sectional curvature of a two-plane \(\pi\) depends only on the angle between \(\pi\) and \(\xi\). In this paper, the authors study several aspects of this class of manifolds. First, they characterize QC-manifolds via an explicit expression for the Riemann curvature tensor. It follows that a QC-manifold is always conformally flat and that the distribution \(\xi^\perp\) is integrable with integral manifolds of constant sectional curvature. Another characterization of QC-manifolds is given using the closely related notion of a biumbilical structure. Second, the authors show that any canal hypersurface in Euclidean space, i.e., the envelope of a one-parameter family of hyperspheres, is a QC-manifold for which the sectional curvature of any two-plane orthogonal to \(\xi\) is strictly positive. Locally, the converse also holds. Next, they consider dilatational transformations of manifolds \((M,g)\) with biumbilical structure \(\xi\). These deform the metric \(g\) in the direction of \(\xi\) only. Some tensorial invariants for this group of transformations are determined. In particular, the QC-property is preserved and, under some weak assumption, the manifold \((M,g,\xi)\) is dilatationally flat if and only if it is a QC-manifold for which the distribution \(\xi^\perp\) has strictly positive sectional curvature. Finally, some applications to Riemannian subprojective manifolds are given.