  We study $N$=2 supersymmetric integrable theories with spontaneously-broken
\Zn\ symmetry. They have exact soliton masses given by the affine $SU(n)$ Toda
masses and fractional fermion numbers given by multiples of $1/n$. The basic
such $N$=2 integrable theory is the $A_n$-type $N$=2 minimal model perturbed by
the most relevant operator. The soliton content and exact S-matrices are
obtained using the Landau-Ginzburg description. We study the thermodynamics of
these theories and calculate the ground-state energies exactly, verifying that
they have the correct conformal limits. We conjecture that the soliton content
and S-matrices in other integrable \Zn\ $N$=2 theories are given by the tensor
product of the above basic $N$=2 \Zn\ scattering theory with various $N$=0
theories. In particular, we consider integrable perturbations of $N$=2
Kazama-Suzuki models described by generalized Chebyshev potentials, $CP^{n-1}$
sigma models, and $N$=2 sine-Gordon and its affine Toda generalizations.
