  We study the non-perturbative dynamics of the two dimensional ${O(N)}$ and
Grassmannian sigma models by using compactification with twisted boundary
conditions on $\mathbb R \times S^1$, semi-classical techniques and resurgence.
While the $O(N)$ model has no instantons for $N>3$, it has (non-instanton)
saddles on $\mathbb R^2$, which we call 2d-saddles. On $\mathbb R \times S^1$,
the resurgent relation between perturbation theory and non-perturbative physics
is encoded in new saddles, which are associated with the affine root system of
the ${\frak o}(N) $ algebra. These events may be viewed as fractionalizations
of the 2d-saddles. The first beta function coefficient, given by the dual
Coxeter number, can then be intepreted as the sum of the multiplicities (dual
Kac labels) of these fractionalized objects. Surprisingly, the new saddles in
$O(N)$ models in compactified space are in one-to-one correspondence with
monopole-instanton saddles in $SO(N)$ gauge theory on $\mathbb R^3 \times S^1$.
The Grassmannian sigma models ${ \rm Gr}(N, M)$ have 2d instantons, which
fractionalize into $N$ kink-instantons. The small circle dynamics of both sigma
models can be described as a dilute gas of the one-events and two-events,
bions. One-events are the leading source of a variety of non-perturbative
effects, and produce the strong scale of the 2d theory in the compactified
theory. We show that in both types of sigma models the neutral bion emulates
the role of IR-renormalons. We also study the topological theta angle
dependence in both the $O(3)$ model and ${ \rm Gr}(N, M)$, and describe the
multi-branched structure of the observables in terms of the theta-angle
dependence of the saddle amplitudes.
