We consider completely positive semidefinite matrices, which are defined as those symmetric matrices that admit a Gram representation by positive semidefinite matrices (of any size). They form the completely positive semidefinite cone, which is nested between the completely positive cone (consisting of the matrices having a Gram representation by nonnegative vectors) and the doubly-nonnegative cone (consisting of the nonnegative matrices having a Gram representation by arbitrary vectors). We will discuss properties of this new matrix cone, hierarchies of inner and outer approximations, and applications in quantum information and convex geometry.