This paper is devoted to the proof of the orbifold theorem: If O is a compact connected orientable irreducible and topologically atoroidal 3-orbifold with nonempty ramiﬁcation locus, then O is geometric (. has a metric of constant curvature or is Seifert ﬁbred). As a corollary, any smooth orientationpreserving nonfree ﬁnite group action on S 3 is conjugate to an orthogonal action. Contents 1. Introduction 2. 3-dimensional orbifolds . Basic deﬁnitions . Spherical and toric decompositions . Finite group actions on spheres with ﬁxed points . Proof of the orbifold theorem from the main theorem 3. .