This paper establishes a rigorous geometrical framework for spherical origami (origami using spherical sheets based on spherical geometry) in two settings: origami restricted to the unit sphere $\mathbb{S}^2$, and three-dimensional folding of spherical sheets in space. Three main results are proved. First, all seven Huzita--Justin axioms admit explicit constructions on $\mathbb{S}^2$, so spherical origami inherits the same one-fold axiom structure as Euclidean origami. Second, the Kawasaki--Justin theorem on flat-foldability extends to $\mathbb{S}^2$ by a local-tangent-plane argument, and the resulting vertex condition is equivalent to the angle-sum condition that characterizes Robertson's spherical $f$-tilings. Third, taking equidistant curves (small arcs) as fold curves yields a one-parameter family of isometric 3D folds, parametrized by the dihedral angle whose closed-form expression is derived, and recovers geodesic folds as the flat limit. The framework is illustrated by constructing computer graphics of spherical origami birds as a concrete example.

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