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Primordial black holes (PBHs) of a wide range of masses could have been formed in the early Universe through different mechanisms. PBHs might explain the dark matter, gravitational-wave events or high-redshift quasars. They can also be used to probe primordial fluctuations on small scales of both scalar and tensor type. The most often discussed mechanism for PBH formation is collapse of small-scale primordial fluctuations during radiation domination. This happens when the amplitude of primordial fluctuations is sufficiently large, and the collapse process, the threshold for PBH formation and their mass can be investigated by numerical simulations. Associated with PBH formation, stochastic gravitational wave backgrounds, CMB spectral distortions, which are deviations from the Planck spectrum of CMB photons, and compact dark matter minihalos may arise. The coalescence of PBHs might be or might have been detected by ongoing and future gravitational wave experiments. Often, the production of PBHs involves fine-tuning, and the degree of fine-tuning depends on primordial non-Gaussianity or formation mechanisms. PBHs might also have been created by high-energy particle collisions in the early Universe, and they could have left over stable relics with their mass given by the fundamental gravity scale instead of fully evaporating by Hawking radiation. In this case such relics can provide the cold dark matter.