The memristor represents the key circuit element for the development of the constitutive blocks of future non-volatile memory architectures and neuromorphic systems. However, resistance switching memories offer a plethora of further opportunities for the electronics of the future. By virtue of the compatibility between the well-established CMOS technology and the fabrication process of most memristors, the exploitation of the peculiar dynamic behaviour of resistance switching memories, which, in general, differ depending upon their material composition, may allow the development of new circuits, which, processing information in unconventional forms, may extend and/or complement the functionalities of state-of-the-art electronic systems. Further, the attractive capability of real-world non-volatile memristors to store and process information in the same physical nanoscale location open the fascinating opportunity to improve the low throughput of Von Neumann computing machines, due to the limited bandwidth of the bus transferring data between the memory and the central processing unit. Finally, the extreme sensitivity of their electrical behaviour to small changes in their initial condition/input and the intrinsic stochastic variability in their switching dynamics may be harnessed to develop innovative bio-signal sensors as well as new cryptographic circuits and systems. The derivation of accurate mathematical models for the electrical behaviour of real-world memristor nano-devices, and their later circuit- and system-theoretic investigation aimed at drawing a comprehensive picture of their peculiar nonlinear dynamic behaviour under the set of inputs and initial conditions expected of the application of interest are fundamental steps towards their conscious future use in integrated circuit design. With this in mind, the present paper adopts a powerful theoretic tool known as Dynamic Route Map to analyse some of the most reliable physics-based models of real-world resistance switching memories to reveal how a particular dynamic phenomenon, known as fading memory, and recently discovered in a tantalum oxide non-volatile memristor device fabricated at Hewlett Packard Labs, is ubiquitous at nanoscale. The physical mechanisms behind the emergence of history-erase effects in non-volatile memristor nano-devices is explained thoroughly for both the DC and the AC periodic excitation scenarios.