The acceleration of high-energy particles at collisionless shocks in astrophysical plasmas is a common phenomenon in the Universe and takes place in a variety of systems, from the heliosphere up to the most distant cosmic sources. The best known example for such non-thermal particle populations are cosmic rays, but additional evidence for the existence of high energy particles comes from observations of non-thermal synchrotron and inverse Compton emission from several sources, as pulsar wind nebulae, jets from active galactic nuclei, gamma-ray bursts, and supernova remnants. In all these sources collisionless shocks are thought to be responsible for the conversion of a significant fraction of the flow energy into relativistic particles with power-law non-thermal spectra. The most popular candidate for particle energization at astrophysical shocks is the so-called diffusive shock acceleration mechanism (first-order Fermi mechanism), where charged particles gain energy by scattering back and forth between the converging upstream and downstream plasmas. In collisionless astrophysical plasmas the source of scattering is provided by magnetic turbulence rather than by Coulomb collisions. Efficient acceleration requires that particles repeatedly cross the shock. For the accelerated particles a power-law spectrum is the natural product of collisionless shock acceleration.I propose to investigate the implications of shock acceleration in gamma-ray bursts and in supernova remnants and to set constraints on the acceleration mechanism. This research project consists of two parts, one concerning the acceleration of electrons at the external shocks of gamma-ray bursts and the resulting emission, and the other one focused on the acceleration of cosmic rays at the front shock of supernova remnants and their diffusion into the Galaxy.