  This review considers the theoretical approaches to the understanding of dark
energy which comprises approximately 68\% of the energy of our Universe and
explains an acceleration in its expansion. Following a discussion of the main
approach based on Einstein's equations with the cosmological term, the
explanations of dark energy using the concept of some kind of scalar field are
elucidated. These include the concept of a quintessence and modifications of
the general theory of relativity by means of the scalar-tensor gravity
exploiting the chameleon, symmetron, and environment-dependent dilaton fields
and corresponding particles. After mentioning several laboratory experiments
allowing to constrain the hypothetical scalar fields modeling the dark energy,
special attention is devoted to the possibility of constraining the parameters
of chameleon, symmetron, and environment-dependent dilaton fields from
measuring the Casimir force. It is concluded that the parameters of each of
these fields can be significantly strengthened in near future by using the next
generation setups in preparation suitable for measuring the Casimir force at
larger separations.
