  For two DG-categories A and B we define the notion of a spherical Morita
quasi-functor A -> B. We construct its associated autoequivalences: the twist T
of D(B) and the co-twist F of D(A). We give powerful sufficiency criteria for a
quasi-functor to be spherical and for the twists associated to a collection of
spherical quasi-functors to braid. Using the framework of DG-enhanced
triangulated categories, we translate all of the above to Fourier-Mukai
transforms between the derived categories of algebraic varieties. This is a
broad generalisation of the results on spherical objects in [ST01] and on
spherical functors in [Ann07]. In fact, this paper replaces [Ann07], which has
a fatal gap in the proof of its main theorem. Though conceptually correct, the
proof was impossible to fix within the framework of triangulated categories.
