We review the theory of spherical twists of derived categories, which are algebraic analogues of Dehn twists around spheres. In particular, we focus on generalized braid group actions generated by spherical twists, which were shown to be faithful for braid groups of type A by Seidel and Thomas. In joint work with Hugh Thomas, we extend such faithfulness results to braid groups of type ADE. Finally, we recall Bridgeland's description of certain spaces of stability conditions associated to a Kleinian singularity, for which our faithfulness result provides the final ingredient.