In three-dimensional anti-de Sitter (AdS) space, there exist several realizations of N-extended supersymmetry, which are traditionally labelled by two non-negative integers p ≥ q such that p + q = N. Different choices of p and q, with N fixed, prove to lead to different restrictions on the target space geometry of supersymmetric nonlinear σ-models. We classify all possible types of hyperkähler target spaces for the cases N = 3 and N = 4 by making use of two different realizations for the most general (p, q) supersymmetric σ-models: (i) off-shell formulations in terms of N = 3 and N = 4 projective supermultiplets; and (ii) on-shell formulations in terms of covariantly chiral scalar superfields in (2,0) AdS superspace. Depending on the type of N = 3, 4 AdS supersymmetry, nonlinear σ-models can support one of the following target space geometries: (i) hyperkähler cones; (ii) non-compact hyperkähler manifolds with a U(1) isometry group which acts non-trivially on the two-sphere of complex structures; (iii) arbitrary hyperkähler manifolds including compact ones. The option (iii) is realized only in the case of critical (4,0) AdS supersymmetry. As an application of the (4,0) AdS techniques developed, we also construct the most general nonlinear σ-model in Minkowski space with a non-centrally extended N = 4 Poincaré supersymmetry. Its target space is a hyperkähler cone (which is characteristic of N = 4 superconformal σ-models), but the σ-model is massive. The Lagrangian includes a positive potential constructed in terms of the homothetic conformal Killing vector the target space is endowed with. This mechanism of mass generation differs from the standard one which corresponds to a σ-model with the ordinary N = 4 Poincaré supersymmetry and which makes use of a tri-holomorphic Killing vector. © 2013 SISSA.