  The main objective of this thesis is to discuss scalar-tensor theories in the
Palatini approach. Both scalar-tensor theories and Palatini formalism are means
of alternating classical theory of gravity, general relativity, in order to
account for phenomena being seemingly unexplainable on the ground of the
Einstein theory or to serve as toy models used to test limitations of the
theory in question. In the literature, both Palatini approach and scalar-tensor
theories have been widely discussed, but there are very few - if none - authors
writing about a merge of these two ideas. The present paper is a result of an
insufficient attention given to the topic of scalar-tensor theories in Palatini
formalism.
  In the course of the thesis action functional for scalar-tensor theories of
gravity will be introduced. This action functional differs significantly from
the action defined in case of scalar-tensor theories in metric approach. We aim
at analysing the theory using the language of invariants, allowing us to write
down all equations in a frame-independent manner. We discover that invariants
defined for the metric case do not always have their counterparts in Palatini
formalism. Also, two frames most frequently used in the literature are
discussed: Einstein and Jordan frame. Possible applications of the theory
developed in the first part of the thesis are presented. We show the
equivalence between $f(R)$ and scalar-tensor theories of gravity and exploit
this fact by analysing the former using methods developed for scalar-tensor
theories. We conclude the thesis with calculating the Friedmann equations for
an empty universe of vanishing spatial curvature and preparing set-up for
analysing inflationary behaviour.
