  Based on the covariant background field method, we calculate the ultraviolet
counter\-terms up to two-loop order and discuss the renormalizability of the
three-dimensional non-linear sigma models with arbitrary Riemannian manifolds
as target spaces. We investigate the bosonic model and its supersymmetric
extension. We show that at the one-loop level these models are renormalizable
and even finite when the manifolds are Ricci-flat. However, at the two-loop
order, we find non-renormalizable counterterms in all cases considered, so the
renormalizability and finiteness of such models are completely lost in this
order.
