The memristor is a new 2-terminal electronic element that complements the classic repertoire of fundamental circuit components consisting of the resistor, inductor and capacitor. Although it was theorised in 1971 by Leon Chua, it remained the object of theoretical interest until 2008 when Hewlett-Packard built, for the first time, novel nanoscale devices whose operation was explained as memristive. Memristors, or “memory-resistors” are nanoscale devices whose memristance depends on the amount of charge that has passed through them. These devices are characterised by extremely useful properties such as low-power consumption, intrinsic non-volatile memory, compatibility with CMOS technology and synapse-like behaviour. Thanks to these properties, they can improve the density and performance of current computer memory technologies by extending the life of CMOS, as well as, enabling hardware implementation of large-scale Artificial Neural Networks (ANN), just to name but a few of their potential applications. Due to their potential impact in these areas, memristive devices have attracted strong attention from the computer industry, with giants such as HP, IBM and Samsung investing heavily in these promising nano-scale devices. The talk will begin with a brief tutorial on what is a memristor and how it behaves. Then it will present an overview of practical memristive devices and how they operate. Finally, we will discuss how the peculiar behaviour and unique properties of memristors can be exploited to improve existing applications or enable new ones.