We develop Möbius sphere geometry for arbitrary euclidean spaces (i.e. real inner product spaces or real pre-Hilbert spaces) X of (finite or infinite) dimension at least 2. All Möbiua transformations of X are determined, especially those which are involutorial. Moreover, M-transformations are characterized within the group of Lie transformations of X. We prove that the 4-point-invariants must be functions of the cross ratio. Stereographic projection from a hypersphere of X ⊕ ℝ onto X U {∞} is introduced, and also Poincaré's model of hyperbolic geometry with respect to an M-ball B and one of the sides σ of B. All bijections of σ preserving hyperbolic distances are determined: they are exactly the Möbius transformations μ satisfying μ (σ) = σ. An isomorphism between the models of Poincaré and Weierstrass of hyperbolic geometry over X is established.