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Astrophysical relativistic jets in active galactic nuclei, gamma-ray bursts, and pulsars is the main key subject of study in the field of high-energy astrophysics, especially regarding the jet interaction with the interstellar or intergalactic environment. In this work, we review studies of particle-in-cell simulations of relativistic electron–proton (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>e</mi><mo>−</mo></msup><mo>−</mo><msup><mi>p</mi><mo>+</mo></msup></mrow></semantics></math></inline-formula>) and electron–positron (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>e</mi><mo>±</mo></msup></semantics></math></inline-formula>) jets, and we compare simulations that we have conducted with the relativistic 3D TRISTAN-MPI code for unmagnetized and magnetized jets. We focus on how the magnetic fields affect the evolution of relativistic jets of different compositions, how the jets interact with the ambient media, how the kinetic instabilities such as the Weibel instability, the kinetic Kelvin–Helmholtz instability and the mushroom instability develop, and we discuss possible particle acceleration mechanisms at reconnection sites.