For a given integer \(d\) more than one, the dynamical moduli space of rational functions of degrees \(d\) is the set of all Mobius conjugacy classes of rational functions of degrees \(d\), which is equipped with both complex analytic and arithmetic structures and has rich properties. In this minicourse, we first recall the foundation of complex dynamics related to stability and bifurcation. Then, based on our recent joint works with Thomas Gauthier and Gabriel Vigny, we will talk about equidistribution in the dynamical moduli space and about improvement of McMullen’s finiteness on the multiplier spectra on the dynamical moduli space, from respectively complex analytic and arithmetic viewpoints.