## 1 Introduction  
The assumption of linear elastic behavior is fundamental in seismic response evaluations, but it may not accurately estimate actual structural damage [1, 2]. More precise analysis methods are needed. Dynamic time history analysis is rigorous but impractical for regular use due to its complexity and the requirement for detailed MDOF models [3]. Nonlinear static pushover analysis, a simpler alternative, is gaining traction for predicting lateral strength and inelastic deformation, although it requires detailed MDOF models [4]. Newer nonlinear algorithms using pushover analysis data have shown promise for more realistic seismic demand estimates and are being integrated into performance-based engineering guidelines [5, 6, 7, 8, 9]. Seismic analysis procedures are categorized into linear and nonlinear methods, each further divided into static and dynamic procedures. The linear static technique is widely used due to its simplicity [10, 11], while linear dynamic techniques offer diverse approaches for studying time-varying structural reactions during seismic events [12, 13]. Figure 1 illustrates the combination of these seismic analysis procedures. Despite the practicality of linear procedures, their accuracy can be limited, especially for complex structures or significant seismic events, necessitating the development of nonlinear procedures [14]. Nonlinear time history (NTH) analyses are the most precise for estimating seismic demands, but their complexity and high computational requirements limit their use in routine engineering applications [15]. Nonlinear stahave been adopted as alternatives and involve monotonously increasing predefined load patterns until a target displacement is reached [16], as recommended in [5]. However, these simplified NSPs can be inaccurate in predicting inelastic seismic demands when higher mode and inelastic effects are significant [16; 17]. To address these limitations, enhanced procedures, such as modal pushover analysis (MPA), modified modal pushover analysis (MMPA) [18], and upper-bound pushover analysis (UBPA) [19], have been developed. In addition, nonlinear procedures, including nonlinear modal response history analysis [20] and nonlinear response history analysis [21], are available. These methods consider the nonlinear behavior of materials and large deformations, but they require significant computational resources [22; 23]. To further improve the prediction of inelastic seismic demands, adaptive pushover procedures were developed. These procedures involve progressively updating load vectors to reflect changes in system modal attributes during the inelastic phase [24]. Additional approaches, such as the adaptive modal combination (AMC) procedure, use adaptive load patterns [16; 25; 26]. These developments represent significant strides towards enhancing the accuracy and practicality of seismic demand estimation. The novelty of this review lies in its comprehensive integration of various pushover methods into a single document, including their procedures, and highlighting their respective advancements. As we navigate through the following sections, we aim to provide an extensive understanding of each method, its advantages and disadvantages, and its applicability in seismic analysis. The subsequent sections are structured as follows:  
* Section 2: an examination of widely adopted nonlinear static analysis techniques. This section represents a comprehensive review of the most utilized nonlinear static analysis procedures, tracing their development and application in seismic studies. * Section 3: an exploration of pushover analysis methods using first modal inertial/ invariant single load vectors. This section represents an in-depth examination of pushover analysis methods that employ first modal inertial or invariant single load vectors, detailing their methodology and significance. * Section 4: a study of pushover analysis with invariant multi-mode vectors.