Nonthermal radiation observed from astrophysical systems containing
(relativistic) jets and shocks, e.g., supernova remnants, active galactic
nuclei (AGNs), gamma-ray bursts (GRBs), and Galactic microquasar systems
usually have power-law emission spectra. Fermi acceleration is the
mechanism usually assumed for the acceleration of particles in
astrophysical environments. Recent PIC simulations using injected
relativistic electron-ion (electro-positron) jets show that accelerationoccurs within the downstream jet, rather than by the scattering of
particles back and forth across the shock as in Fermi acceleration. Shock
acceleration is a ubiquitous phenomenon in astrophysical plasmas. Plasma
waves and their associated instabilities (e.g., the Buneman
instability, other two-streaming instability, and the Weibel
instability) created in the shocks are responsible for particle
(electron, positron, and ion) acceleration. The simulation results show
that the Weibel instability is responsible for generating and amplifying
highly nonuniform, small-scalemagnetic fields. These magnetic fields contribute to the electron's
transverse deflection behind the jet head. The \"jitter'' radiation from
deflected electrons has different properties than synchrotron radiation
which is calculated in a uniform magnetic field. This jitter radiation may
be important tounderstanding the complex time evolution and/or spectral structure in
gamma-ray bursts, relativistic jets, and supernova remnants. We will
review recent PIC simulations which show particle acceleration in jets.