The sphere SX = {x∈X: 〈x, x〉 = 1} of a right Hilbert C*-module X over a unital C*-algebra B is studied using differential geometric techniques. An action of the unitary group of the algebra [Lscr ]B(X) of adjointable B-module operators of X makes SX a homogeneous space of this group. A reductive structure is introduced, as well as a Finsler metric. Metric properties of the geodesic curves are established. In the case B a von Neumann algebra and X self-dual, the fundamental group of SX is computed.