Recall that two finitely generated groups G and H are quasi-isometric, if they admit a topological coupling, i.e. an action of G times H on a locally compact topological space such that each factor acts properly and cocompactly. This topological definition of quasi-isometry was given by Gromov, and at the same time he proposed a measure theoretic analogue of this definition, called the measure equivalence, which is closely related to the notion of orbit equivalence in ergodic theory. Despite the similarity in the definition of measure equivalence and quasi-isometry, their relationship is rather mysterious and not well-understood. We study the relation between these two notions in the class of right-angled Artin groups. In this talk, we show if H is a countable group with bounded torsion which is integrable measure equivalence to a right-angled Artin group G with finite outer automorphism group, then H is finitely generated, and H and G are quasi-isometric. This allows us to deduce integrable measure equivalence rigidity results from the relevant quasi-isometric rigidity results for a large class of right-angled Artin groups. Interestly, this class of groups are rigid for a reason which is quite different from other cases of measure equivalence rigidity. We will also do a quick survey of relevant measure equivalence rigidity and quasi-isometric rigidity results of other classes of groups, motivating our choice of right-angled Artin groups as a playground. This is joint work with Camille Horbez.