We discuss physical processes underlying the generation of ion beams with high emission current density in electrohydrodynamic (EHD) emitters based on liquid metals and alloys and with room temperature ionic liquids. We consider EHD effects that influence the emission of ions (ion production mechanisms) and the kinetics of ion interactions in high-density beams. We analyze the factors determining the emission zone size, sustainability of emission at high and low currents, generation of clusters, increase in energy spread, decrease in brightness, and other features of ion beams. We consider the specific design features of EHD emitters and the problems of practically ensuring their stable operation. Discussed in detail are modern application areas for ion sources with EHD emitters, including technological installations for ion beam lithography, micro- and nanopatterning, ion microscopes and tools for local mass spectrometry of secondary ions, and systems to control and neutralize the potential of spacecraft as well as electrostatic rocket engines (microthrusters). We analyze prospects for further development of EHD emitters themselves and instruments based on them.