The manipulation of liquid crystals by external potentials, such as electric or magnetic fields, is of great importance in many industrial and research applications. To gain insight into the impact of the different mechanisms in such a complex multi-particle system is a scientific challenge. Interactions between particles and external potential, particle-particle interactions, and, in case of colloidal systems, hydrodynamic interactions lead, in their interplay, to fascinating dynamical states and unusual diffusion behavior. This work presents new insights into the dynamics of colloidal liquid crystals with computer simulations. For this purpose several model systems of increasing complexity are studied. The first model system is the most basic model system possible: a system of hard spherocylinders that only interact via excluded volume. In a next step Brownian motion, the random motion of colloidal particles in a fluid, is included via Langevin Dynamics. Finally, also the impact of hydrodynamic interactions between the particles is studied within a Lattice Boltzmann framework.