One of the mathematical cornerstones of modern data ana- lytics is machine learning whereby we automatically learn subtle patterns which may be hidden in training data, we associate those patterns with outcomes and we apply these patterns to new and unseen data and make predictions about as yet unseen outcomes. This form of data analytics al- lows us to bring value to the huge volumes of data that is collected from people, from the environment, from commerce, from online activities, from scientific experiments, from many other sources. The mathematical basis for this form of machine learning has led to tools like Support Vector Machines which have shown moderate effectiveness and good efficiency in their implementation. Recently, however, these have been usurped by the emergence of deep learning based on convolutional neural networks. In this presentation we will examine the basis for why such deep net- works are remarkably successful and accurate, their similarity to ways in which the human brain is organised, and the challenges of implementing such deep networks on conventional computer architectures.