Lifetime data are often subject to complicated censoring mechanisms. In particular, point inspection schedules result in observations for which the exact failure times are known only to fall in an interval. Furthermore, overlapping intervals occur when more than one inspection schedule is employed. While well-known parametric and nonparametric inference procedures exist, the piecewise exponential (PEX) model provides a flexible alternative. The PEX model is characterized by a piecewise-constant hazard function with specified jump points. The jump points may be determined as a function of the data, giving the model a nonparametric interpretation, or according to physical considerations related to the process but independent of the data. Assumptions concerning the shape of the hazard function can be incorporated into the model;The EM algorithm provides a useful method of estimation, particularly as the number of hazard jump points increases. Its convergence is guaranteed even when the MLE lies on the boundary of the parameter space. A version of the EM algorithm is used to construct approximate confidence intervals based on inverting the likelihood ratio test statistic. Asymptotic properties of the PEX estimator are given for certain censoring mechanisms. A Monte Carlo study was done to investigate the effect of a constrained hazard function and of the choice of jump points on the resulting estimate of the survival function. The performance of the likelihood ratio based confidence intervals is also evaluated.