A straightforward, cheap and unique method to produce novel fibers with a diameter in the range of 100 nm and even less is related to electrospinning. For this goal, polymer solutions, liquid crystals, suspensions of solid particles and emulsions, are electrospun in the electric field of about 1 kV/cm. The electric force results in an electrically charged jet of polymer solution flowing out from a pendant or sessile droplet. After the jet flows away from the droplet in a nearly straight line, it bends into a complex path and other changes in shape occur, during which electrical forces stretch and thin it by very large ratios. After the solvent evaporates, birefringent nanofibers are left. Nanofibers of ordinary, conducting and photosensitive polymers were electrospun. The present review deals with the mechanism and electrohydrodynamic modeling of the instabilities and related processes resulting in electrospinning of nanofibers. Also some applications are discussed. In particular, a unique electrostatic field-assisted assembly technique was developed with the aim to position and align individual conducting and light-emitting nanofibers in arrays and ropes. These structures are of potential interest in the development of novel polymer-based light-emitting diodes (LED), diodes, transistors, photonic crystals and flexible photocells. Some other applications discussed include micro-aerodynamic decelerators and tiny flying objects based on permeable nanofiber mats (smart dust), nanofiber-based filters, protective clothing, biomedical applications including wound dressings, drug delivery systems based on nanotubes, the design of solar sails, light sails and mirrors for use in space, the application of pesticides to plants, structural elements in artificial organs, reinforced composites, as well as nanofibers reinforced by carbon nanotubes.