We prove global rigidity for compact hyperbolic and spherical cone-3-manifolds with cone-angles ≤ π (which are not Seifert
fibered in the spherical case), furthermore for a class of hyperbolic cone-3-manifolds of finite volume with cone-angles ≤ π, possibly
with boundary consisting of totally geodesic hyperbolic turnovers.To that end we first generalize the local rigidity result contained
in [Wei] to the setting of hyperbolic cone-3-manifolds of finite volume as above. We then use the techniques developed in [BLP] to
deform the cone-manifold structure to a complete non-singular or a geometric orbifold structure, where global rigidity holds due to
Mostow-Prasad rigidity, cf. [Mos], [Pra], in the hyperbolic case,resp. [deR], cf. also [Rot], in the spherical case. This strategy
has already been implemented successfully by [Koj] in the compact hyperbolic case if the singular locus is a link using HodgsonKerckhoff local rigidity, cf. [HK].