  A search for the $P$- and $CP(T)$-violating electric dipole moments (EDM) of
atoms, particles and nuclei with sensitivity up to $10^{-15}$ in units of
magnetic dipole moments, allowed by all discrete symmetries, is one of the
topical problems of modern physics. According to Sakharov, $CP$-violation is
one of the three key criteria of baryogenesis in generally accepted paradigm of
the Big Bang cosmology. All three criteria are supported by the Standard Model
(SM), but it fails to describe the observed baryon asymmetry of the Universe.
This is regarded a strong argument in favor of existence of $CP$-symmetry
breaking mechanisms beyond minimal SM, which can lead to measurable EDMs of
atoms, particles and nuclei. Direct searches for EDM of charged particles and
nuclei are possible only in storage rings (COSY, NICA). After successful
studies by the JEDI collaboration at the COSY synchrotron, at the forefront is
a search for the proton EDM in an electrostatic storage ring with the proton
spin frozen at the magic energy with projected sensitivity $d_p\sim
10^{-29}\,e\cdot$cm. Following a brief introduction to the $CP$-violation
physics and the baryogenesis, the review presents a detailed discussion of
significant contributions to the spin dynamics from the terrestrial gravity
along with the new effects of Earth's rotation in ultrasensitive searches for
the EDM of charged particles and neutrons. Quite remarkably, for the projected
sensitivity to the proton EDM, these false EDM effects can by one to two orders
of magnitude exceed the signal of the proton EDM, and become comparable to an
EDM contribution in experiments with ultracold neutrons. We also discuss the
role of a precessing spin as a detector of the axion-like dark matter, and
consider applications of the quantum gravitational anomalies to the dense
matter hydrodynamics and the spin phenomena in the non-central nuclear
collisions.
