The cyclins comprise a family of proteins which combine with protein kinase subunits encoded by members of the cdc2 family. The active protein kinase thus formed phosphorylates target proteins in the cell and promotes certain cell cycle transitions. Cyclins A and B show the unusual property of sudden and specific proteolysis shortly before the metaphase-anaphase transition during mitosis. The destruction of the cyclins leads to rapid loss of activity of their kinase companions. The 'simple idea' referred to in the title of this article was that accumulation of cyclin formed maturation promoting factor (MPF), the enzyme that promotes the G2-M transition, and cyclin destruction turned off MPF. The complexity refers to additional controls of cdc2 activity, such as its reversible phosphorylation. Furthermore, many new members of the cyclin family have recently been discovered that may play a role in the regulation of the G1-S transition, and in higher organisms, the cdc2 family is also more numerous than was at first appreciated. Cyclins make important contributions both to subcellular localization and the substrate specificity of their companion kinase subunits. It is too early to say if the entire range of cyclin-like and cdc2-like protein kinases are involved in the control of the cell cycle.