Chiral symmetry and its anomalous and spontaneous breaking play an important role

in particle physics, where it explains the origin of pion and hadron mass hierarchy

among other things.

Despite its microscopic origin chirality may also lead to observable effects

in macroscopic physical systems -- relativistic plasmas made of chiral

(spin-$\frac{1}{2}$) particles.

Such plasmas are called \textit{chiral}.

The effects include non-dissipative currents in external fields that could be present

even in quasi-equilibrium, such as the chiral magnetic (CME) and separation (CSE)

effects, as well as a number of inherently chiral collective modes

called the chiral magnetic (CMW) and vortical (CVW) waves.

Applications of chiral plasmas are truly interdisciplinary, ranging from

hot plasma filling the early Universe, to dense matter in neutron stars,

to electronic band structures in Dirac and Weyl semimetals, to quark-gluon plasma

produced in heavy-ion collisions.

The main focus of this dissertation is a search for traces of chiral physics

in the spectrum of collective modes in chiral plasmas.

I start from relativistic chiral kinetic theory and derive

first- and second-order chiral hydrodynamics.

Then I establish key features of an equilibrium state that describes many

physical chiral systems and use it to find the full spectrum of collective modes

in high-temperature and high-density cases.

Finally, I consider in detail the fate of the two inherently chiral waves, namely

the CMW and the CVW, and determine their detection prospects.

The main results of this dissertation are the formulation of a fully covariant

dissipative chiral hydrodynamics and the calculation of the spectrum of collective

modes in chiral plasmas.

It is found that the dissipative effects and dynamical electromagnetism play

an important role in most cases.

In particular, it is found that both the CMW and the CVW are heavily damped by the usual

Ohmic dissipation in charged plasmas and the diffusion effects in neutral plasmas.

These findings prompt a search for new physical observables in heavy-ion collisions,

as well as a revision of potential applications of chiral theories in

cosmology and solid-state physics.