The NPMLE of a distribution function under random one-sided (left or right) truncation is the well-known Lynden-Bell estimator. However, in some applications, left and right truncation occur simultaneously, and Lynden-Bell is inconsistent. In Survival Analysis and Epidemiology, this double truncation is typically encountered when the recruited inter-event times correspond to terminating events (e.g. cancer or AIDS diagnosis) falling between two fixed dates. Another interesting example of random double truncation is the problem of two-sided detection limits for quasar luminosities in Astronomy. Under double truncation, the NPMLE has no explicit form, and its computation and the derivation of its statistical properties are far from obvious. In this talk I will review the Efron-Petrosian NPMLE and a semiparametric counterpart which is based on a parametric specification of the joint distribution of the (random) truncation limits. Relative advantages and disadvantages will be discussed, as well as the existing software for the computation of the estimators. Time permitting, I will also consider the problem of estimating the underlying density. Extensive simulations and applications to epidemiological and astronomical data will be provided. This is joint work with Carla Moreira (University of Vigo) and Rosa Crujeiras (University of Santiago de Compostela).