Electric power systems are critical infrastructures subject to the possibility of extraordinary events with high societal consequences. Although such possibilities are often associated with very low probabilities of being realized, it is nevertheless crucial to be able to identify and understand the vulnerabilities of power systems related to extraordinary events. The objective of the work presented in this article is to establish a methodological basis for vulnerability analysis that is complementary to conventional risk and reliability analysis of power systems. It presents a comprehensive framework of definitions, indicators and methods that can be used to classify, analyse and monitor vulnerabilities in power transmission and distribution systems. Its main components include 1) a conceptual framework of definitions that forms the basis for understanding and classifying vulnerabilities, 2) an assessment methodology for identifying vulnerabilities related to extraordinary events and barriers to mitigate them, and 3) vulnerability indicators for quantifying and monitoring power system vulnerabilities. The applicability of the vulnerability analysis framework is demonstrated through several studies of real power systems. Moreover, the concept of power system vulnerability elaborated in this article is also related to the concept of power system resilience.