The failure of a power system as a critical infrastructure causes considerable damage to society. Hence, the vulnerabilities of such facilities should be minimized to cope with several sources of disruption. Various methods have been proposed to identify and address the weaknesses of power systems to enhance their robustness and resilience. As the field is evolving quickly, understanding the pros and cons of each approach and the trends could be challenging. This paper aims to guide the reader toward choosing the most effective method according to the issue investigated. We focus on studies on power grids; however, research on other critical infrastructure could also benefit from this review. We identified three classes of events, namely natural hazards, intentional attacks, and random failures. These events affect the adopted method that can range from analytical approaches—complex network, flow-based, logical, and functional methods—to Monte Carlo simulations. At present, hybrid approaches are emerging with the growing complexities of power grids. Various methods are used in combination to benefit from the strengths of one another. We identified three emerging topics and challenges that require further investigations, namely the N-k problem, trade-off between robustness and optimality, and emerging drivers in power grids.