An overview of general phenomenology and proposed physical mechanisms of large scale electrical discharges termed 'blue jets' and 'gigantic jets' observed at high altitude in the Earth's atmosphere above thunderstorms is presented. The primary emphasis is placed on summarizing available experimental data on the observed morphological features of upward jet discharges and on the discussion of recently advanced theories describing electrodynamic conditions, which facilitate escape of conventional lightning leaders from thundercloud tops and their upward propagation toward the ionosphere. It is argued that the filamentary plasma structures observed in blue jet and gigantic jet discharges are directly linked to the processes in streamer zones of lightning leaders, scaled by a significant reduction of air pressure at high altitudes.