Several chronic diseases are characterized by an initial asymptomatic stage during which, if detected by screening, they can be cured in a more effective way. This article considers two statistical design problems in screening for chronic disease: the choice of examination ages and the choice of the part of the population to be screened. One main goal is capturing the trade-off between the costs of examination and the losses due to late detection, while accounting for the effects of age on the incidence of the disease, on mortality, and on the relative advantages of early detection. The problem is posed in a decision theoretic way. The model adopted considers a single individual, whose history relative to the disease is represented by a discrete-valued stochastic process. The transition structure is general, but known. The decision space includes all sequences of examination times, as well as no examination. The optimality criterion accounts for the cost of examinations and, in a general way, for the goals of screening in terms of mortality and morbidity. So the optimality criterion may depend on survival, quality-adjusted life years, cost of care, and so on, as well as on combinations of these factors. A general solution and computational algorithms are derived by extending to this context methodologies developed in reliability theory. The case in which the test used for screening has high sensitivity is studied in detail; then the determination of the optimal schedule and stopping rule is reduced to a one-dimensional optimization problem by recursive dynamic methods. Moreover, sufficient conditions for screening to be increasingly worthwhile with age are derived. Under these conditions, the optimal number of planned examinations is either 0 or infinity, and there is a simple check to establish whether or not to screen without having to compute the optimal schedule. Under slightly stronger conditions, the times between examinations decrease and the optimal solution is unique and easy to compute. The conditions mentioned relate increasing times between checks to properties of the failure rate of the time to onset of the disease and of the relative incidence of the disease. Applications of the results include developing guidelines for screening for breast and cervical cancers—currently a controversial issue.