In this paper we review some recent approaches that have been used to address the difficult problem of estimating the unknown size of a finite population.We start from illustrating what types of inferential difficulties one should expect when no parametric assumption is made on the class of distributions for the distribution

of counts of the number of multiple occurrences of the same unit when the observed counts are modelled in terms of Poisson mixtures. We then consider the problem from the species sampling model perspective where each unit is represented by the distinct species and a sequence of exchangeable observations are available. We discuss the implementation of the alternative approaches with real datasets and we compare their performance with simulated data.