The classical symmetric space Q_n of positive definite inner products on R^n admits a properly discontinuous action of GL(n,Z), and the topology of the quotient can be used to study properties of this group. A more geometric description of Q_n is as the space of flat n-tori together with a basis for the fundamental group. By analogy with Q_n, Culler-Vogtmann outer space CV_n parametrizes rank n, metric graphs together with a choice of free basis for their fundamental group. CV_n has been an indispensable tool for understanding the outer automorphism groups of free groups. Right-angled Artin groups (raags) are a well-studied family of groups containing both free groups and free abelian groups, which have recently featured prominently in low-dimensional topology. For each raag A we construct a finite-dimensional outer space, parametrizing certain CAT(0) cube complexes equipped with a cocompact action of A. This is joint work with Ruth Charney and Karen Vogtmann.