  The main subjects of the PhD dissertation concern cosmological models
considered in Palatini f(R) gravity and scalar-tensor theories. We introduce a
simple generalization of the LCDM model based on Palatini modified gravity with
quadratic Starobinsky term. A matter source is provided by generalized
Chaplygin gas. The statistical analysis of our model is investigated. We use
dynamical system approach to study the evolution of the Universe. The model
reaches a very good agreement with the newest experimental data and yields an
inflationary epoch caused by a singularity of the type III. The present-day
accelerated expansion is also provided by the model. We also show that the Lie
and Noether symmetry approaches are very useful tools in cosmological
considerations. We examine two other models of Extended Theories of Gravity
(ETGs): the novel hybrid metric-Palatini gravity and a minimally coupled to
gravity scalar field. The first one is applied to homogeneous and isotropic
model while in the scalar-tensor theory we study anisotropic universes. We use
the symmetries in order to find unknown forms of potential and to solve
classical field equations in both models. The symmetries also are very helpful
in searching exact and invariant solutions of Wheeler-DeWitt equations. In the
last part we are interested in equilibrium configurations and stability
conditions of relativistic stars in the framework of scalar-tensor theories.
Firstly, we show that TOV-like form of the equilibrium equations can be
obtained for a wide class of ETGs if generalized energy density and pressure
are defined. According to our studies, a neutron star is a stable system for
the minimally coupled scalar field model. There is a supplement including notes
on symmetries as well as dynamical systems approach. The illustrative examples
of applications are also provided.
