Heterogenous nucleation is not understood in more detail than homogenousnucleation. E.g., in the case of hard colloidal spheres a flat wall increases the rate of crystalgrowth by orders of magnitude compared to homogenous nucleation, while a rough wallcan suppress the formation of crystal completely for a certain roughness of the wall withrespect to the size of the particles. The transition from surface properties that promote or suppress heterogeneous nucleation is not understood in detail. Thus, it is the main goal of this project to reach a detailed understanding of heterogeneous crystal nucleation close to flat, curved, rough, and structured surfaces that can either accelerate or suppress nucleation. We will focus on a deeper understanding on the microscopic level of single particles, as the properties of crystal precursors and critical crystal nuclei are of prime interest for gaining a better understanding of the nucleation process.