The basic building blocks of every neural network are neurons and their inter-cellular connections, called synapses. In nature, synapses play a crucial role in learning and memory, since they are plastic, which means that they change their state depending on the neural activity of the respectively coupled neurons. In neuromorphic systems, the functionality of neurons and synapses is emulated in hardware systems by employing very-large-scale integration technology. In this context, it seems rather natural to use non-volatile memory technology to mimic synaptic functionality. In particular, memristive devices are promising candidates for neuromorphic computing, since they allow one to emulate synaptic functionalities in a detailed way with a significantly reduced power usage and a high packing density. This tutorial aims to provide insight on current investigations in the field to address the following fundamental questions: How can functionalities of synapses be emulated with memristive devices? What are the basic requirements to realize artificial inorganic neurons and synapses? Which material systems and device structures can be used for this purpose? And how can cellular synaptic functionality be used in networks for neuromorphic computing? Even if those questions are part of current research and not yet answered in detail, our aim is to present concepts that address those questions. Furthermore, this tutorial focuses on spiking neural models, which enables mimicking biological computing as realistically as possible. Published by AIP Publishing.