Among the emerging memories, resistive switching (RS) based resistive random-access memories (RRAMs) are attracting lots of attention due to their simple metal–insulator–metal structures, low power consumption, long endurance and retention characteristics, low fabrication cost, ultrafast switching, and CMOS compatibility. In recent years, several oxides, chalcogenides, polymers and their composites have been explored for RS devices. Many of these studies show a high resistance ratio of the OFF and ON states with good RS characteristics. Some of these studies also show the realization of multifunctional RS devices such as the simultaneous switching of resistance and magnetic states. In order to scale up RS-based RRAMs, a detailed understanding of the occurrence of RS is very much desirable. In this review, we provide an overview of the current understanding, recent advances and future outlook of RS-based RRAM devices along with fundamental concepts of the different types of RS, and conventional as well as novel measurement techniques which are being used to characterize RS devices. Observations of RS in different materials are presented, and RS mechanisms, such as the valence change mechanism and electrochemical metallization memory, are discussed in detail. An overview of multifunctional RS devices and the main challenges faced in scaling up RS devices is also presented.