Motivated by the search for solvable string theories, we consider the problem of classifying the integrable bosonic 2d sigma-models. We include non-conformal sigma-models, which have historically been a good arena for discovering integrable models that were later generalized to Weyl-invariant ones. General sigma-models feature a quantum RG flow, given by a 'generalized Ricci flow' of the target-space geometry.

This thesis is based on the conjecture that integrable sigma-models are renormalizable, or stable under the RG flow. It is widely understood that classically integrable theories are stable at the leading 1-loop order with only a few parameters running.

Here we address what happens at higher-loop orders. We find that integrable sigma-models generally remain RG-stable at higher-loops provided they receive a particular choice of finite counterterms, or quantum (alpha') corrections to the target-space geometry. Thought to be preserving integrability at the quantum level, these corrections are analogous to those required for higher-loop conformal invariance of gauged Wess-Zumino-Witten models.

We explicitly construct the quantum corrections restoring higher-loop renormalizability for examples of integrable eta- and lambda-deformed sigma-models. We then consider the integrable GxG and GxG/H models and also construct a new class of integrable GxG/H models with abelian H.

We also reformulate the lambda-models as sigma-models on a "tripled" GxGxG configuration space, where they become automatically renormalizable due to manifest global symmetries. In the limit when they become non-abelian dual (NAD) models, this suggests that the corresponding 'interpolating models' for NAD are also renormalizable, with 2-loop beta-functions matching the group/symmetric space models.

We then present a new and different link between integrability and the RG flow in the context of sigma-models with 'local couplings' depending explicitly on 2d time. Such models are naturally obtained in the light-cone gauge in string theory, pointing to the possibility of a large, new class of solvable string models.