Populations of cells are almost always heterogeneous in function and fate. To understand the plasticity of cells, it is vital to measure quantitatively and dynamically the molecular processes that underlie cell-fate decisions in single cells. Early events in cell signalling often occur within seconds of the stimulus, whereas intracellular signalling processes and transcriptional changes can take minutes or hours. By contrast, cell-fate decisions, such as whether a cell divides, differentiates or dies, can take many hours or days. Multiparameter experimental and computational methods that integrate quantitative measurement and mathematical simulation of these noisy and complex processes are required to understand the highly dynamic mechanisms that control cell plasticity and fate.