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In the modern era, artificial intelligence (AI) and deep learning (DL) seamlessly integrate into various spheres of our daily lives. These cutting-edge disciplines have given rise to numerous safety-critical applications such as autonomous driving with a paramount concern on ensuring a high promise of dependability because of the high risk of human injury in the case of malfunction. Even the dependability becomes more crucial as shrinking CMOS technology feature size enhances resilience concerns due to factors like aging. In the context of DL accelerators, which heavily rely on the efficiency and speed of computations, addressing the effects of aging is of utmost significance to ensure their optimal design and performance. This paper addresses the overarching dependability issue of advanced deep neural networks (DNN) accelerators from the aging perspective. Especially, a comprehensive survey and taxonomy of techniques used to evaluate and mitigate aging effects are introduced. We cover different aging effects like permanent faults, timing errors, and lifetime issues. We review research by the layer-wise approach and categorize several resilience classes to bring out major features. The concluding part of this review highlights the questions answered and several future research directions. This study is expected to benefit researchers in different areas of DNN deployment, especially the dependability of this emergent paradigm.