This document explores the mathematics of deep convolutional neural networks. We begin at the level of an individual neuron, and from there examine parameter tuning, fully-connected networks, error minimization, back-propagation, convolutional networks, and finally deep networks. The report concludes with experiments on geometric invariance, and data augmentation. Relevant MATLAB code is provided throughout, and a downloadable package is available at the end of the document. Artificial neural networks (ANNs) [1] are at the core of state-of-the-art approaches to a variety of visual recognition tasks, including image classification [2] and object detection [3]. For a computer vision researcher interested in recognition, it is useful to understand how ANNs work, and why they have recently become so effective. An artificial neural network is a type of biologically-inspired pattern recognizer.