In transportation planning, one of the first steps is to estimate the travel demand. The final product of the estimation process is an origin-destination (OD) matrix, whose entries correspond to the number of trips between pairs of origin-destination zones in a study region. In this paper, we review the main statistical models proposed in the literature for the estimation of the OD matrix based on traffic counts. Unlike reconstruction models, statistical models do not aim at estimating the exact OD matrix corresponding to observed traffic volumes, but they rather aim at estimating the parameters of a statistical model of the population of OD matrices. Initially we define the estimation problem, emphasizing its underspecified nature, which has lead to the development of several models based on different approaches. We describe static models whose parameters are estimated by means of maximum likelihood, the method of moments, and Bayesian inference. We also describe  some recent dynamic models. Following that, we discuss research questions related to the underspecification problem, model assumptions and the estimation of the route choice matrix, and indicate promising research directions.