  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 sigma-models. We classify all possible
types of hyperkahler target spaces for the cases N=3 and N=4 by making use of
two different realizations for the most general (p,q) supersymmetric
sigma-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 sigma-models can support one of the following target
space geometries: (i) hyperkahler cones; (ii) non-compact hyperkahler manifolds
with a U(1) isometry group which acts non-trivially on the two-sphere of
complex structures; (iii) arbitrary hyperkahler 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 sigma-model in Minkowski space with a non-centrally
extended N=4 Poincare supersymmetry. Its target space is a hyperkahler cone
(which is characteristic of N=4 superconformal sigma-models), but the
sigma-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 sigma-model with the ordinary N=4 Poincare
supersymmetry and which makes use of a tri-holomorphic Killing vector.
