Ultra-high intensity (UHI) laser shots (i.e. with a peak intensity greater than 10^18 W.cm^-2) are able to ionize gaseous or solid targets in mere femtoseconds, and in the meantime, accelerate electrons up to relativistic energies (ranging from the MeV to the GeV). The ensuing huge electron currents give rise, via various mechanisms, to a plethora of particle- (ions, neutrons, etc.) and photon- (gamma, X, XUV, THz) secondary sources whose unique properties (duration, compactness, flux, etc.) render them well suited to diverse applications (target imaging, spectroscopy, nuclear physics, etc.). UHI lasers also make it possible to produce and analyze very high energy density- or extremely magnetized matter samples, to study the fundamental processes underlying relativistic plasmas, or even, to recreate in the lab astrophysical phenomena which rest simultaneously upon collective and quantum electrodynamics processes. All these topics are studied theoretically and numerically using the particle-in-cell simulation code CALDER running on the high-performance computing facilities at CEA. These studies also contribute to the design and interpretation of experiments carried out in other laboratories.