This paper presents a CGAL kernel for algorithms manipulating 3D spheres, circles, and circular arcs. The paper makes three contributions. First, the design of the kernel concept is developed, following the best practices for the design of kernels geared towards curved objects. Second, we show how two different frameworks can be combined: one for the general setting, and one dedicated to the case where all the objects handled are located on a reference sphere. In both cases, the mathematical derivations for predicates and constructions are detailed. Third, an application to the construction of the exact arrangement of circles on a sphere is overviewed.