Asignificant rise in Artificial Intelligence (AI) has impacted many applications around us, so much so that AI has now been increasingly used in safety-critical applications. AI at the edge is the reality, which means performing the data computation closer to the source of the data, as opposed to performing it on the cloud. Safety-critical applications have strict reliability requirements; therefore, it is essential that AI models running on the edge (i.e., hardware) must fulfill the required safety standards. In the vast field of AI, Deep Neural Networks (DNNs) are the focal point of this survey as it has continued to produce extraordinary outcomes in various applications i.e. medical, automotive, aerospace, defense, etc. Traditional reliability techniques for DNNs implementation are not always practical, as they fail to exploit the unique characteristics of the DNNs. Furthermore, it is also essential to understand the targeted edge hardware because the impact of the faults can be different in ASICs and FPGAs. Therefore, in this survey, first, we have examined the impact of the fault in ASICs and FPGAs, and then we seek to provide a glimpse of the recent progress made towards the fault-tolerant DNNs. We have discussed several factors that can impact the reliability of the DNNs. Further, we have extended this discussion to shed light on many state-of-the-art fault mitigation techniques for DNNs.