This paper proposes a new way of modeling age, period, and cohort effects that improves substantively and methodologically on the conventional linear model. The linear model suffers from a well-known identification problem: If we assume an outcome of interest depends on the sum of an age effect, a period effect, and a cohort effect, then it is impossible to distinguish these three separate effects because, for any individual, birth year = current year – age. Less well appreciated is that the model also suffers from a conceptual problem: It assumes that the influence of age is the same in all time periods, the influence of present conditions is the same for people of all ages, and cohorts do not change over time. We argue that in many applications, these assumptions fail. We propose a more general model in which age profiles can change over time and period effects can have different influences on people of different ages. Our model defines cohort effects as an accumulation of age-by-period interactions. Although a long-standing literature on theories of social change conceptualizes cohort effects in exactly this way, we are the first to show how to statistically model this more complex form of cohort effects. We show that the additive model is a special case of our model and that, except in special cases, the parameters of the more general model are identified. We apply our model to analyze changes in age-specific mortality rates in Sweden over the past 150 years. Our model fits the data dramatically better than the additive model. The estimates show that the rate of increase of mortality with age among adults became more steep from 1881 to 1941, but since then the rate of increase has been roughly constant. The estimates also allow us to test whether early-life conditions have lasting impacts on mortality, as under the cohort morbidity phenotype hypothesis. The results give limited support to this hypothesis: The impact of early-life conditions lasts for several years but is unlikely to reach all the way to old age.