We present three original results for the dynamics of rational maps on the Riemann sphere. Using methods from dimension and ergodic theory, we discuss generalized physical measures and prove their existence for hyperbolic and some parabolic rational maps. This shows that there are sets of typical points for these maps having maximal dimension. We then show that for any NCP rational map, the set of non-typical, or divergence, points is also of maximal dimension. Finally, we examine holomorphic families of stable rational maps and show that the dimension of the maps depends plurisubharmonically on the parameters.