Abstract We discuss quantization of SO( N + 1) σ models and CP N models, and of certain non-compact counterparts, SO( N , 1) and QP N respectively, of these, both in canonical operator formalism and the covariant path integral formulation, showing the equivalence of the two approaches. We discuss also a class of supersymmetric σ models formulated in d ⩽ 3 dimensions and apply the results to the SO( N + 1) and SO( N , 1) cases. This allows us to calculate the Witten index in each case. For SO(2 l + 1,1) we thereby find supersymmetry breaking. However, for SO(2 l , 1), we find supersymmetry is unbroken. Moreover, there is no unique ground state, invariant under SO(2 l , 1), rather an infinite multiple of zero energy states, carrying a unitary irreducible representation of the non-compact SO(2 l , 1) group. We discuss also field theoretic aspects of the models in d ⩾ 2 dimensions, stressing differences of the non-compact to the compact cases. These include infrared instead of ultraviolet asymptotic freedom, lack of an energy gap, failure (in the QP N case) of the auxiliary vector field to become dynamical. A further conclusion that is argued concerns the absence of a consistent particle interpretation for the QP N model in exactly two dimensions. For d > 2 the non-compact symmetry of QP N is broken down to the compact subgroup.