There is significant interest in studying stem cells, both to elucidate their basic biological functions during development and adulthood as well as to learn how to utilize them as new sources of specialized cells for tissue repair. Whether the motivation is basic biology or biomedical application, however, progress will hinge upon learning how to better control stem-cell function at a quantitative and molecular level. There are several major challenges within the field, including the identification of new signals and conditions that regulate and influence cell function, and the application of this information towards the design of stem-cell bioprocesses and therapies. Both of these efforts can significantly benefit from the synthesis of biological data into quantitative and increasingly mechanistic models that not only describe, but also predict, how a stem cell's environment can control its fate. This review will briefly summarize the history and current state of the stem-cell biology field, but will then focus on the development of predictive models for stem-cell control. Early models formulated on the assumption that cell fate was decided by stochastic, cell-intrinsic processes have gradually evolved into hybrid deterministic-stochastic models with increasingly finer molecular resolution that accounts for environmental regulation. As our understanding of cellular control mechanisms expands from the cell surface and towards the nucleus, these efforts may culminate in the development of a stem-cell culture programme, or a series of signals to provide to the cells as a function of time to guide them along a desired developmental trajectory.