Memory is the cornerstone of intelligence. State-of-the-art artificial intelligence (AI) systems based on artificial neural networks (NNs) are no exception. They are powered by two core "memory algorithms": (1) learning algorithms that translate past experience into model parameters/programs, and (2) sequence-processing neural architectures that maintain and manipulate task-relevant information while solving a problem at hand. Together with internet-scale data, advances in these memory systems have enabled major breakthroughs in real-world AI applications, including increasingly general-purpose chatbots. Despite this progress, today's best AI systems still lack several cognitive capabilities associated with truly general or superintelligent machines---I argue that these limitations stem largely from the shortcomings of the existing memory algorithms. In this talk, I will present "memory-centric deep learning", a research programme focused on developing advanced memory algorithms to address key frontier challenges in AI. I will further show how progress in machine learning algorithms and models can also contribute to advancing the study of natural intelligence by providing computational models and insights that are not available in the traditional toolbox of cognitive neuroscience. Overall, this research programme aims to advance an integrated science of intelligence---one that unites machine intelligence and natural cognition, and creates new opportunities for multidisciplinary collaboration.