Importance sampling is a very well-known variance-reducing technique used in Monte Carlo simulations of radiative transport. It involves a distortion of the physical (analog) transition probabilities with the goal of causing events of interest in the computation to occur more frequently than in the analog process. This distortion is then compensated by a corresponding alteration of the estimating random variable in order to remove any bias from the estimates of quantities of interest. In this paper, we construct several families of estimators based on importance sampling methods to solve general transport problems and prove that the adaptive application of each estimator produces geometric convergence of the approximate solution. We also present numerical results that illustrate important elements of the theory.