Fragility analysis aims to compute the probabilities of a system exceeding certain damage conditions given different levels of hazard intensity. Fragility analysis is therefore a key process of performance-based earthquake engineering, with a number of approaches developed and widely recognized, including Incremental Dynamic Analysis (IDA), Multiple Stripe Analysis (MSA), and cloud analysis. Additionally, extended fragility analysis has recently been shown to possess important attributes of mathematical consistency and extensibility. This work provides a critical review of the different fragility methods by explaining the underlying probabilistic models and assumptions, as well as their connections to the extended fragility method. It is proven that IDA-based fragility curves provide an upper bound of the actual fragility, and cloud analysis manifests suboptimality issues arising from its underlying assumptions. MSA is identified to be a probit-linked Bernoulli regression model, similar to the one proposed by Shinozuka and coworkers. The latter, in turn, is shown to be a limiting subcase of the generalized linear model framework introduced within the extended fragility analysis. The paper first presents a simple case of one intensity measure and two damage condition states, and the discussion is subsequently extended to more general cases of multiple intensity measures and damage states. The discussed attributes are demonstrated in several numerical applications. Overall, this work aims to provide new insights on fragility methods, enabling efficient, accurate, and consistent estimations of structural performance, as well as promoting new research directions in earthquake engineering and other related fields.@en