Long ago, Achúcarro and Townsend discovered that in three dimensions (3D)-extended anti-de Sitter (AdS) supergravity exists in several incarnations, which were called the (p, q) AdS supergravity theories with non-negative integers p≥ q such that. Using the superspace approach to 3D-extended supergravity developed in arXiv: 1101.4013, we present three superfield formulations for supergravity that allow for well defined cosmological terms and supersymmetric AdS solutions. The conformal compensators corresponding to these theories are respectively:(i) a chiral scalar multiplet;(ii) a vector multiplet; and (iii) an improved complex linear multiplet. The theories corresponding to (i) and (iii) are shown to provide two dually equivalent realizations of the (1, 1) AdS supergravity, while (ii) describes the (2, 0) AdS supergravity. We associate with each supergravity formulation, with and without a cosmological term, a consistent super-current multiplet. The supercurrents in the (1, 1) and (2, 0) AdS backgrounds are derived for the first time. We elaborate on rigid supersymmetric theories in (1, 1) and (2, 0) AdS superspaces.