  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.
  We explicitly construct these quantum corrections for examples of integrable
$\eta$- and $\lambda$-deformed $\sigma$-models. We then reformulate the
$\lambda$-models as $\sigma$-models on a "tripled" $G \times G \times G$
configuration space, where they become automatically renormalizable due to
manifest symmetries and a decoupling of some fields. We also consider the
integrable $G \times G$ and $G \times G/H$ models and construct a new class of
integrable $G \times G/H$ models with abelian $H$.
  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.
