This article provides a brief introduction to the Faraday Discussion "New memory paradigms: memristive phenomena and neuromorphic applications" held in Aachen, Germany, 15–17 October 2018. It will cover basic definitions of memristive switching elements, their main switching modes, and their most important performance parameters as well as applications in neuromorphic computing. The article comprises parts from the following sources: General Introduction and Introduction to Part V of Nanoelectronics and Information Technology, ed. R. Waser, Wiley-VCH, 2012; Chapter 4 of Nanotechnology: Volume 3: Information Technology I, ed. R. Waser, Wiley-VCH, Weinheim, 2008; Chapters 3–9 of Emerging Nanoelectronic Devices, ed. A. Chen, J. Hutchby, V. Zhirnov and G. Bourianoff, Wiley, 2015; Chapter 1 of Resistive Switching, ed. D. Ielmini and R. Waser, Wiley-VCH, 2016 (with permission by Wiley-VCH).