Cosmological shocks convert a fraction of the energy of gravitationally accelerated flows to internal energy of the gas. They heat and compress the gas and can also accelerate energetic non-thermal particles and amplify magnetic fields. We discussed some specific features of cosmological shocks.
• The standard Rankine-Hugoniot relations based on the conservation laws for a steady single-fluid MHD shock allow to calculate the state of the fluid behind the shock once the upstream state and the shock strength are known. The coplanarity theorem for a plane ideal MHD shock states that the upstream and downstream bulk velocities, magnetic fields and the shock normal all lie in the same plane.
• Cosmological plasma shocks are likely to be collisionless as many other astrophysical shocks observed in the heliosphere and in supernova remnants. We review the basic plasma processes responsible for the microscopic structure of collisionless shocks.
• Collisionless shock heating of ions results in a non-equilibrium state just behind a very thin magnetic ramp region with a strongly anisotropic quasi-Maxwellian ion distributions. The possibility of collisionless heating of electrons by electromagnetic fluctuations in the magnetic ramp region depends on the extension of the fluctuation spectra to the electron gyro-scales, and could depend on the shock Mach number. Then the Coulomb equilibration processes are operating on the scales much larger than the collisionless shock width.
• Extended MHD shock waves propagating in turbulent media
could accelerate energetic particles both by Fermi type
acceleration in converging plasma flows and by DC electric field in
quasi-perpendicular shocks. If the acceleration is efficient, then
the strong shock could convert a substantial fraction (more than
10%) of the power dissipated by the upstream bulk flow to energetic
particles (cosmic rays). The compression ratio rtot at
such a shock can be much higher, while the ion temperature behind the
shock 
rtot-2 and the post-shock entropy are
lower, than that in a standard single fluid shock. The shock
structure consists of an extended precursor and a viscous velocity
jump (subshock) indicated in Fig. 5.
• Strong collisionless plasma shocks with an efficient Fermi
acceleration of energetic particles could generate strong MHD waves
in the upstream and downstream regions and strongly amplify the
upstream magnetic fields. A distinctive feature of the shock is a
predicted possibility of gas pre-heating in the far upstream region
due to MHD wave dissipation, that can produce an extended filament
of temperature 
0.1 keV.
• Shock waves both from the cosmic web formation processes and those due to cluster merging activity can play an important role in clusters of galaxies. Direct evidences for such shocks, as traced by radio relics and the temperature jumps in X-ray observations havebeen found only in a small number of clusters, and thus we need more observations.
Acknowledgements. The authors thank ISSI (Bern) for support of the team "Non-virialized X-ray components in clusters of galaxies". A.M.B. thanks M.Yu. Gustov for his help with hybrid shock simulations. He acknowledges the RBRF grant 06-02-16844, a support from RAS Presidium Programmes. A support from NASA ATP (NNX07AG79G) is acknowledged.