Postulated by Wolfgang Pauli in 1925 to explain the electronic structure of atoms and molecules, the exclusion principle establishes an upper bound of 1 for the fermionic natural occupation numbers (ni ≤ 1), thereby allowing no more than one electron in each quantum state. This is nothing more than a necessary and sufficient condition for a 1-body reduced density matrix to be the contraction of an N-body density matrix, provided that the sum of such occupations equals the number of particles. The pure N-representability problem asks whether given natural occupation numbers can arise from an antisymmetric pure N-particle state. Recent progress on such a problem establishes a systematic approach to a set of stronger constraints on the natural occupation numbers, not implied by the exclusion principle. In fact, for a pure quantum system the natural occupations satisfy a set of linear inequalities, known as generalized Pauli constraints, of the form with . This generalization of the Pauli exclusion principle has been widely used in many-body physics to study the influence of the fermionic exchange symmetry, beyond the original Pauli principle. In this chapter we review recent applications of the generalized Pauli exclusion principle and discuss some of the open problems.