A major goal in complex dynamics is to understand dynamical moduli spaces; that is, conjugacy classes of holomorphic dynamical systems. One of the great successes in this regard is the study of the moduli space of quadratic polynomials; it is isomorphic to the complex plane. This moduli space contains the famous Mandelbrot set, which has been extensively studied over the past 40 years. Understanding other dynamical moduli spaces to the same extent tends to be more challenging as they are often higher-dimensional. Many tools from complex analysis that pave the way for key breakthroughs in the one-dimensional setting do not carry over to higher dimensions. So instead of considering the whole moduli space, we follow an approach initiated by William Thurston and investigate special subvarieties of moduli space that give rise to dynamical moduli spaces. In this talk, we will explore the topology and geometry of the dynamical moduli spaces that play a prominent role in complex dynamics.