A magnetic tunnel junction (MTJ), which consists of a thin insulating layer (a tunnel barrier) sandwiched between two ferromagnetic electrode layers, exhibits tunnel magnetoresistance (TMR) due to spin-dependent electron tunnelling. Since the 1995 discovery of room-temperature TMR, MTJs with an amorphous aluminium oxide (Al-O) tunnel barrier have been studied extensively. Al-O-based MTJs exhibit magnetoresistance (MR) ratios up to about 70% at room temperature (RT) and are currently used in magnetoresistive random access memory (MRAM) and the read heads of hard disk drives. MTJs with MR ratios significantly higher than 70% at RT, however, are needed for next-generation spintronic devices. In 2001 first-principle theories predicted that the MR ratios of epitaxial Fe/MgO/Fe MTJs with a crystalline MgO(001) barrier would be over 1000% because of the coherent tunnelling of fully spin-polarized Delta(1) electrons. In 2004 MR ratios of about 200% were obtained in MTJs with a single-crystal MgO(001) barrier or a textured MgO(001) barrier. CoFeB/MgO/CoFeB MTJs for practical applications were also developed and found to have MR ratios up to 500% at RT. MgO-based MTJs are of great importance not only for device applications but also for clarifying the physics of spin-dependent tunnelling. In this article we introduce recent studies on physics and applications of the giant TMR in MgO-based MTJs.