Colloidal particles grafted with single-stranded DNA (ssDNA) chains can self-assemble into a number of different crystalline structures, where hybridization of the ssDNA chains creates links between colloids stabilizing their structure. Depending on the geometry and the size of the particles, the grafting density of the ssDNA chains, and the length and choice of DNA sequences, a number of different crystalline structures can be fabricated. However, understanding how these factors contribute synergistically to the self-assembly process of DNA-functionalized nano- or micro-sized particles remains an intensive field of research. Moreover, the fabrication of long-range structures due to kinetic bottlenecks in the self-assembly are additional challenges. Here, we discuss the most recent advances from theory and experiment with particular focus put on recent simulation studies.