Long ago, Achucarro and Townsend discovered that in three dimensions (3D)N-extended anti-de Sitter (AdS) supergravity exists in several incarnations,which were called the (p,q) AdS supergravity theories with non-negativeintegers p>=q such that N=p+q. Using the superspace approach to 3D N-extendedsupergravity developed in arXiv:1101.4013, we present three superfieldformulations for N=2 supergravity that allow for well defined cosmologicalterms and supersymmetric AdS solutions. The conformal compensatorscorresponding to these theories are respectively: (i) a chiral scalarmultiplet: (ii) a vector multiplet; and (iii) an improved complex linearmultiplet. The theories corresponding to (i) and (iii) are shown to provide twodually equivalent realizations of the (1,1) AdS supergravity, while (ii)describes the (2,0) AdS supergravity. We associate with each supergravityformulation, with and without a cosmological term, a consistent supercurrentmultiplet. The supercurrents in the (1,1) and (2,0) AdS backgrounds are derivedfor the first time. We elaborate on rigid supersymmetric theories in (1,1) and(2,0) AdS superspaces.