Eukaryotic checkpoint controls impose delays in the cell cycle in response to DNA damage or defects in DNA replication. Genetic and physiological studies in budding yeast have identified key genes and defined genetic pathways involved in checkpoint-mediated responses. Recent studies now lead to biochemical models that explain at least in part the arrest in G1 and delays during DNA replication after damage. Though progress in checkpoint controls has indeed been rapid, several observations identify puzzling aspects of checkpoint controls with few plausible explanations.