Intermediate-mass black holes (IMBHs), with masses in the range $100-10^{6}$ M$_{odot}$, are the link between stellar-mass BHs and supermassive BHs (SMBHs). They are thought to be the seeds from which SMBHs grow, which would explain the existence of quasars with BH masses of up to 10$^{10}$ M$_{odot}$ when the Universe was only 0.8 Gyr old. The detection and study of IMBHs has thus strong implications for understanding how SMBHs form and grow, which is ultimately linked to galaxy formation and growth, as well as for studies of the universality of BH accretion or the epoch of reionisation. Proving the existence of seed BHs in the early Universe is not yet feasible with the current instrumentation; however, those seeds that did not grow into SMBHs can be found as IMBHs in the nearby Universe. In this review I summarize the different scenarios proposed for the formation of IMBHs and gather all the observational evidence for the few hundreds of nearby IMBH candidates found in dwarf galaxies, globular clusters, and ultraluminous X-ray sources, as well as the possible discovery of a few seed BHs at high redshift. I discuss some of their properties, such as X-ray weakness and location in the BH mass scaling relations, and the possibility to discover IMBHs through high velocity clouds, tidal disruption events, gravitational waves, or accretion disks in active galactic nuclei. I finalize with the prospects for the detection of IMBHs with up-coming observatories.
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