Spin-transfer-torque magnetoresistive random-access memory (STT-MRAM) is an emerging nonvolatile memory that uses magnetic tunnel junctions (MTJs) to store information. MTJs with a crystalline MgO(001) tunnel barrier sandwiched between ferromagnetic layers, such as CoFeB, exhibit giant tunnel magnetoresistance, which is used to readout the STT-MRAM. Writing of STT-MRAM is based on current-induced magnetization reversal, called STT switching. STT-MRAM with perpendicular magnetization is especially important for high-density and low-power-consuming memory applications such as embedded memory for large-scale integrated circuit. For STT-MRAM to replace ultrahigh-density dynamic random-access memory, however, there are still technological challenges concerning the materials and fabrication processes of MTJs. This article reviews the physics and materials science of MTJs for STT-MRAM. We also discuss the importance of new MTJ materials and processes for next-generation ultrahigh-density MRAM.