The brain is immensely complex, with diverse components and dynamic interactions building upon one another to orchestrate a wide range of behaviors. Understanding the patterns of these complex interactions and how they are coordinated to support collective neural function is critical for parsing human and animal behavior, treating mental illness, and developing artificial intelligence. Rapid experimental advances in imaging, recording, and perturbing neural systems across various species now provide opportunities to distill underlying principles of brain organization and function. Here, we take stock of recent progress and review methods used in the statistical analysis of brain networks, drawing from fields of statistical physics, network theory, and information theory. Our discussion is organized by scale, starting with models of individual neurons and extending to large-scale networks mapped across brain regions. We then examine organizing principles and constraints that shape the biological structure and function of neural circuits. We conclude with an overview of several critical frontiers, including expanding current models, fostering tighter feedback between theory and experiment, and leveraging perturbative approaches to understand neural systems. Alongside these efforts, we highlight the importance of contextualizing their contributions by linking them to formal accounts of explanation and causation.