  The Standard Model of elementary particles and their interactions does not
include the gravitational interaction and faces problems in understanding of
the dark matter, dark energy, strong CP violation etc. In continuing attempts
to solve these problems, many predictions of new light elementary particles and
hypothetical interactions beyond the Standard Model have been made. These
predictions can be constrained by many means and, specifically, by measuring
the Casimir force arising between two closely spaced bodies due to the
zero-point and thermal fluctuations of the electromagnetic field. After a brief
survey in the theory of the Casimir effect, the strongest constraints on the
power-type and Yukawa-type corrections to Newtonian gravity, following from
measuring the Casimir force at short distances, are considered. Next,the
problems of dark matter, dark energy and their probable constituents are
discussed. This is followed by an analysis of constraints on the dark matter
particles, and, specifically, on axions and axionlike particles, obtained from
the Casimir effect. The question of whether the Casimir effect can be used for
constraining the spin-dependent interactions is also considered. Then the
constraints on the dark energy particles, like chameleons and symmetrons, are
examined. In all cases the subject of our treatment is not only measurements of
the Casimir force but some other relevant table-top experiments as well. In
conclusion, the prospects of the Casimir effect for constraining theoretical
predictions beyond the Standard Model at short distances are summarized.
