Deep Learning (DL) applications are gaining increasing interest in the industry and academia for their outstanding computational capabilities. Indeed, they have found successful applications in various areas and domains such as avionics, robotics, automotive, medical wearable devices, gaming; some have been labeled as safety-critical, as system failures can compromise human life. Consequently, DL reliability is becoming a growing concern, and efficient reliability assessment approaches are required to meet safety constraints. This paper presents a survey of the main DL reliability assessment methodologies, focusing mainly on Fault Injection (FI) techniques used to evaluate the DL resilience. The article describes some of the most representative state-of-the-art academic and industrial works describing FI methodologies at different levels of abstraction. Finally, a discussion of the advantages and disadvantages of each methodology is proposed to provide valuable guidelines for carrying out safety analyses.