A mathematical model is developed, detailing the manner in which radiation brings about the transformation of cells to a state of uncontrolled growth. The model is based on the concepts of initiation and promotion, with the irradiation acting both to damage intracellular structures and to change the state of cells surrounding a damaged (initiated) cell. The complete model requires that the radiation produce two forms of damage within a cell, with at least one of the forms requiring an interaction which is a function of time since irradiation. Some form of contact inhibition must be removed, with this step being a function of the probability that a cell in an initiated state will be surrounded by n dead cells. The cell then must divide, with the probability of moving the cell to the final transformed state being a function of the number of cellular divisions. Prior to irradiation, it is assumed that cells may be characterized by an initial state vector describing the probability that any given cell is in one of the states specified by the model. The resulting model then is used to explain data concerning in vitro irradiation of cells by acute doses of X-rays, alpha particles and neutrons. Limited tests of the theory under conditions of fractionated irradiation are also provided. A controlling factor in such studies is the number of cells already in intermediate states prior to the irradiation.