Discrete-time survival data typically possess three features: discreteness, ties, and concomitant information, which require appropriate discrete-time models to analyze. In this paper, we first review some existing discrete-time survival models and then extend them to discrete-time cure survival models, which account for the presence of long-term survivors (cured individuals). The maximum likelihood estimation as well as approximate partial likelihood approaches are used to estimate the model parameters. Simulation results are shown to support the suitability of such models for discrete-time survival data with long-term survivors. An example of applications on a set of bladder tumor recurrence data is also presented.