The cumulant expansion gives rise to an useful decomposition of the two-matrix in which the pair correlated matrix (cumulant) is disconnected from the antisymmetric product of the one-matrices. The cumulant can be approximated in terms of two matrices, Δ and Π, which are explicit functions of the occupation numbers of the natural orbitals. It produces a natural orbital functional (NOF) that reduces to the exact expression for the total energy in two-electron systems. The N-representability positivity necessary conditions of the two-matrix impose several bounds on the matrices Δ and Π. Appropriate forms of these matrices lead to different implementations of the NOF known in the literature as PNOFi (i = 1–5). The basic features of these functionals are reviewed here. The strengths and weaknesses of the different PNOFs are assessed. © 2012 Wiley Periodicals, Inc.