One goal of researchers in neuroscience, psychology, and artificial intelligence is to build theoretical models that are able to explain the flexibility and adaptiveness of biological systems. How to build a brain provides a detailed guided exploration of a new cognitive architecture that takes biological detail seriously, while addressing cognitive phenomena. The Semantic Pointer Architecture (SPA) introduced in this book provides a set of tools for constructing a wide range of biologically constrained perceptual, cognitive, and motor models. Examples of such models are provided, and they are shown to explain a wide range of data including single cell recordings, neural population activity, reaction times, error rates, choice behavior, and fMRI signals. Each of these models introduces a major feature of biological cognition addressed in the book, including semantics, syntax, control, learning, and memory. These models are not introduced as independent considerations of brain function, but instead integrated to give rise to what is currently the world's largest functional brain model. The last half of this book compares the Semantic Pointer Architecture with the current state-of-the-art, addressing issues of theory construction in the behavioral sciences, semantic compositionality, and scalability, among other considerations. The book concludes with a discussion of conceptual challenges raised by this architecture, and identifies several outstanding challenges for this, and other, cognitive architectures. Along the way, the book considers neural coding, concept representation, neural dynamics, working memory, neuroanatomy, reinforcement learning, and spike-timing dependent plasticity. The book includes 8 detailed, hands-on tutorials exploiting the free Nengo neural simulation environment, providing practical experience with the concepts and models presented throughout.