  We consider the Chiral Cosmological Models (CCMs) and modified gravity
theories associated with them. Generalization of the superpotential method for
a general CCM with several scalar fields is performed, and the method of
construction CCMs admitting exact solutions is developed. New classes of exact
solutions in the two-component CCM connected with an $f(R)$ gravity model with
an additional scalar field have been constructed. We construct new cosmological
solutions for a diagonal metric of the target space, including modified
power-law solutions. In particular, we propose the reconstruction procedure
based on the superpotential method and present examples of kinetic part
reconstruction for periodic and hyperbolic Hubble parameters. We also focus on
a cyclic type of Universe dubbed the Quasi-Steady State (QSS) model, with the
aim of constructing single- and double-field potentials for one and the same
behaviour of the Hubble parameter using the developed superpotential method for
the CCM. The realization of this task includes a new set of solutions for a CCM
with a scale factor characterized by the QSS theory. We also propose a method
for reducing the two-field CCM to the single scalar field model.
