As deep learning systems are scaled up to many billions of parameters, relating their internal structure to external behaviors becomes very challenging. Although daunting, this problem is not new: Neuroscientists and cognitive scientists have accumulated decades of experience analyzing a particularly complex system - the brain. In this work, we argue that interpreting both biological and artificial neural systems requires analyzing those systems at multiple levels of analysis, with different analytic tools for each level. We first lay out a joint grand challenge among scientists who study the brain and who study artificial neural networks: understanding how distributed neural mechanisms give rise to complex cognition and behavior. We then present a series of analytical tools that can be used to analyze biological and artificial neural systems, organizing those tools according to Marr's three levels of analysis: computation/behavior, algorithm/representation, and implementation. Overall, the multilevel interpretability framework provides a principled way to tackle neural system complexity; links structure, computation, and behavior; clarifies assumptions and research priorities at each level; and paves the way toward a unified effort for understanding intelligent systems, may they be biological or artificial.