There is an extensive literature on statistical modeling and estimation for a univariate failure time variate T > 0. In this chapter some core methods, which we will build upon in subsequent multivariate failure time methods presentations, will be described. The core methods include Kaplan–Meier survivor function estimation, Cox model hazard ratio parameter estimation with its associated logrank test, as well as other censored data rank tests. The presentation will focus on nonparametric and semiparametric likelihood formulations for estimator development for reasons mentioned in Chapter 1. A brief account of asymptotic distribution theory for these testing and estimation procedures will also be given.