  The Standard Model (SM) is the best description of fundamental particles and
their interactions we have to date. From this theory, all phenomena in the
macroscopic world (except for gravity) can be explained, and it has
successfully predicted all outcomes of particle experiments on Earth. However,
cosmological observations of the early Universe yield a large imbalance between
its content of matter and antimatter, which is several orders of magnitude
above the SM prediction. To explain these observations, new interactions that
do not respect the charge-parity symmetry must exist beyond the SM. Such
interactions would induce electric dipole moments (EDMs) in known particles.
  In Part I of this thesis, we propose to extend the active experimental
program of EDM searches to charm and bottom baryons, $\tau$ leptons, and
$\Lambda$ hyperons; also allowing the measurement of their corresponding
magnetic dipole moments (MDMs). The EDM and MDM of short-lived particles can be
accessed with a bent-crystal experiment to be installed in the Large Hadron
Collider (LHC), while longer-lived $\Lambda$ particles can be measured at the
LHCb experiment with no additional instrumentation. In Part II, an analysis of
LHCb data to measure the $\Lambda$ polarization in $\Lambda_c^+ \to \Lambda
\pi^+ \pi^+ \pi^-$ decays, essential ingredient for the proposed measurement,
is presented. In the last part of the thesis, we derive new indirect bounds on
heavy quark EDMs with already available data and explore the phenomenological
implications of these and other EDM limits on New Physics models, with special
emphasis on extensions of the SM with colour-octet scalars.
