Until recently, the only technique to prove representability results in rigid geometry went via the theory of formal models. This suffices to construct, e.g., the Quot/Hilb spaces in sufficient generality, but not the Picard space pararmetrizing invertible line bundles on a proper rigid variety. By proving and using a version of Artin's representability criteria in the rigid setting in addition to the formal methods, I will construct this Picard space under certain hypotheses. Key to this development is the thoroughgoing use of the theory of adic spaces.