Phase Transitions In and Out of Equilibrium

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This PhD thesis is subdivided into two parts. In the first part we study the dynamics of phase separation on the surface of a spherical particle. By connecting our findings with studies of patchy particles, we consider the implications on self-assembly of many-particle systems. We provide extended information on dynamical density functional theory, the Gaussian Core model and the numerical methods for computations on the surface of a sphere. In the second part we investigate the isotropic-nematic transition of self-propelled rods in three spatial dimensions. We precisely describe the Brownian dynamics simulation method and its numerical implementation for ellipsoidal Gay-Berne particles.