Intergalactic electromagnetic cascades in the magnetized Universe as a tool of astroparticle physics

We review the physics of intergalactic electromagnetic cascades in the presence of the extragalactic magnetic field (EGMF). Various regimes of intergalactic electromagnetic cascades are considered depending on the number of cascade generations, the value of the cascade electron deflection angle, and the relations between the EGMF coherence length, typical cascade $\gamma$-ray mean free path, and electron energy loss length. We also review contemporary constraints on the EGMF parameters and explore the sensitivity of various $\gamma$-ray instruments to the EGMF parameters.