Let $$M \in \mathbb {R}^{p \times q}$$ M ∈ R p × q be a nonnegative matrix. The positive semidefinite rank (psd rank) of M is the smallest integer k for which there exist positive semidefinite matrices $$A_i, B_j$$ A i , B j of size $$k \times k$$ k × k such that $$M_{ij} = {{\mathrm{trace}}}(A_i B_j)$$ M i j = trace ( A i B j ) . The psd rank has many appealing geometric interpretations, including semidefinite representations of polyhedra and information-theoretic applications. In this paper we develop and survey the main mathematical properties of psd rank, including its geometry, relationships with other rank notions, and computational and algorithmic aspects.