Proper progression through the cell-division cycle is critical to normal development and homeostasis and is necessarily misregulated in cancer. The key to cell-cycle regulation is the control of two waves of transcription that occur at the onset of DNA replication (S phase) and mitosis (M phase). MuvB complexes play a central role in the regulation of these genes. When cells are not actively dividing, the MuvB complex DREAM represses G1/S and G2/M genes. Remarkably, MuvB also forms activator complexes together with the oncogenic transcription factors B-MYB and FOXM1 that are required for the expression of the mitotic genes in G2/M. Despite this essential role in the control of cell division and the relationship to cancer, it has been unclear how MuvB complexes inhibit and stimulate gene expression. Here we review recent discoveries of MuvB structure and molecular interactions, including with nucleosomes and other chromatin-binding proteins, which have led to the first mechanistic models for the biochemical function of MuvB complexes.