  Cosmological hydrodynamic simulations have demonstrated that shock waves
could be produced in the intergalactic medium by supersonic flow motions during
the course of hierarchical clustering of the large-scale-structure in the
Universe. Similar to interplanetary shocks and supernova remnants (SNRs), these
structure formation shocks can accelerate cosmic ray (CR) protons and electrons
via diffusive shock acceleration. External accretion shocks, which form in the
outermost surfaces of nonlinear structures, are as strong as SNR shocks and
could be potential accelerations sites for high energy CR protons up to
$10^{18}$ eV. But it could be difficult to detect their signatures due to
extremely low kinetic energy flux associated with those accretion shocks. On
the other hand, radiative features of internal shocks in the hot intracluster
medium have been identified as temperature and density discontinuities in X-ray
observations and diffuse radio emission from accelerated CR electrons. However,
the non-detection of gamma-ray emission from galaxy clusters due to $\pi^0$
decay still remains to be an outstanding problem.
