Magnetic random access memory (MRAM) possesses the attractive properties of non-volatility, radiation hardness, nondestructive readout, low voltage, high access speed, unlimited read and write endurance and high density. MRAM technology is described for the devices using giant magnetoresistance (GMR) and tunneling magnetoresistance (TMR) materials in this paper. The TMR type MRAM architectures using ferromagnetic tunneling junctions (MTJ) are more attractive for mainstream RAM applications than the GMR type, because the signal of the TMR type is larger than that of the GMR type. A MRAM device with an MTJ plus MOS transistor switch architecture, which can provide large signal-to noise ratio, is detailed. A design of the MTJ element is discussed and the requirements for the junction resistance and the TMR needed for the memory device are demonstrated based on the simple signal voltage calculations. The TMR significantly decreases with increasing bias voltage, which leads to the reduction of the signal voltage for the actual MRAM. A ferromagnetic double tunneling junction is proposed for the high density MRAM application demanding large signal voltage, because of the smaller degradation of the TMR for the bias voltage, compared with that of the conventional single junctions. Recent trials of MRAM fabrication are introduced, which demonstrates high-speed access time. Finally, challenges for the higher bit density MRAM above Gb are discussed, and it is noticed that higher signal voltage, lower power consumption for writing and novel cell designs are needed for the achievement.