The article reviews the current state of studies of electrical breakdown and discharges in vacuum. The main attention is devoted to cathode and near-cathode phenomena, which form the physical content of fast processes in a vacuum discharge. The model of cathode initiation of a vacuum discharge as the result of field emission with high current density is analyzed. The phenomenon of explosive electron emission is described in detail. On the basis of ideas developed in study of explosive electron emission, nonstationary processes in vacuum breakdown and in the cathode spot of a vacuum arc are discussed.