This thesis contains contributions to the theory of spherical convex bodies, i.e., closed convex sets on the unit n-sphere. On one hand projection covariant binary operations on the set of spherical convex bodies are investigated. In Euclidean convexity Minkowski addition, a projection covariant binary operation, together with the volume gives rise to the Brunn-Minkowski theory. This theory lies at the very core of classical Euclidean convexity and provides a unifying framework for various extremal and uniqueness problems for convex bodies in R^n. However, in spherical convexity there is no known natural analogue to Minkowski addition. Together with Franz Schuster, all projection covariant binary operations on the set of spherical convex bodies contained in a fixed open hemisphere are characterized and it is shown, that the convex hull is essentially the only non-trivial projection covariant binary operations between pairs of convex bodies contained in open hemispheres. On the other hand a spherical analogue of the Euclidean convex floating body is introduced and investigated. Together with Elisabeth Werner, a new notion of spherical convex floating bodies is defined and the volume difference of a spherical convex body and its floating body is investigated. Remarkably, this volume difference gives rise to a new spherical area measure, the floating area. This floating area can be seen as a spherical analogue of the classical affine surface area from affine differential geometry. We start an investigation of the properties of this new spherical quantity.