Truncation occurs when the variable of interest can be observed only if its value satisfies certain selection criteria. Most existing methods for analyzing such data critically rely on the assumption that the truncation variable is quasi-independent of the variable of interest. In this article, the authors propose a likelihood-based inference approach under the assumption that the dependence structure of the two variables follows a general form of copula model. They develop a model selection method for choosing the best-fitted copula among a broad class of model alternatives, and they derive large-sample properties of the proposed estimators, including the inverse Fisher information matrix. The treatment of ties is also discussed. They apply their methods to the analysis of a transfusion-related AIDS data set and compare the results with existing methods. Simulation results are also provided to evaluate the finite-sample performances of all the competing methods.