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The last decade of machine learning has seen drastic increases in scale and capabilities. Deep neural networks (DNNs) are increasingly being deployed in the real world. However, they are difficult to analyze, raising concerns about using them without a rigorous understanding of how they function. Effective tools for interpreting them will be important for building more trustworthy AI by helping to identify problems, fix bugs, and improve basic understanding. In particular, "inner" interpretability techniques, which focus on explaining the internal components of DNNs, are well-suited for developing a mechanistic understanding, guiding manual modifications, and reverse engineering solutions. Much recent work has focused on DNN interpretability, and rapid progress has thus far made a thorough systematization of methods difficult. In this survey, we review over 300 works with a focus on inner interpretability tools. We introduce a taxonomy that classifies methods by what part of the network they help to explain (weights, neurons, subnetworks, or latent representations) and whether they are implemented during (intrinsic) or after (post hoc) training. To our knowledge, we are also the first to survey a number of connections between interpretability research and work in adversarial robustness, continual learning, modularity, network compression, and studying the human visual system. We discuss key challenges and argue that the status quo in interpretability research is largely unproductive. Finally, we highlight the importance of future work that emphasizes diagnostics, debugging, adversaries, and benchmarking in order to make interpretability tools more useful to engineers in practical applications.