The Random Neural Network (RNN) is a biologically inspired neural network model proposed by Prof. Gelenbe in which signals are represented as discrete spikes. I will start by describing certain properties of neuronal signalling in real biological neurons and how these have inspired the RNN model. Next, I will introduce the mathematics of RNN and will provide selected examples of RNN applications. These are modelling of biological neural systems, texture recognition and segmentation, image and video compression and multicast routing.