## 1 Introduction  
Masonry is a widespread material that has been employed for centuries in various types of structures around the world, ranging from monuments and historical construction to ordinary buildings like houses, many of them settled in earthquake-prone areas. Unreinforced masonry (URM) buildings, while capable of withstanding gravitational loads, are highly vulnerable to horizontal dynamic loads. Therefore, existing URM buildings, including historical buildings and monuments, are susceptible to damage involving cracks or the partial or total collapse of the structure, even under moderate ground motion. Masonry is a composite material made of units and joints. Units comprise a wide range of materials, such as natural stones, bricks, adobe, and others. Joints can be dry or made of cement-, lime-, or mud-based mortar. The poor performance of unreinforced masonry under lateral loads, particularly seismic excitations, arises from the low tensile and shear strength of the material. The damage of masonry during earthquakes are normally classified into two types: in-plane failure and out-of-plane failure. The three basic in-plane (IP) damage patterns are flexural failure, shear-diagonal failure, and shear sliding (see Figure 1). This type of response is more ductile [1]. Geometry, boundaries, loading conditions, and mechanical properties govern the load capacity and failure mode. On the other hand, openings affect the IP behaviour creating points of weakness [2]. In the case of out-of-plane (OOP) damage, vertical, horizontal, or diagonal (or a combination of) flexural cracks develop, resulting in partial or complete collapse [3], as shown in Figure 1. This type of damage is fragile, being the first collapse mechanism and more frequent in existing URM buildings when no prevention is taken [4]. OOP mechanisms require less energy compared to IP mechanisms for activation. This implies that even a small seismic action can induce OOP damage or even lead to collapses [5,6]. Consequently, OOP mechanisms have a higher risk for both the stability of the building and human life. OOP mechanisms are more likely to occur when there are weak connections with other structural elements, such as transversal walls, floors, and roofs. Moreover, flexible floors and roofs can contribute to the collapse of walls in the OOP direction. The geometrical layout (in elevation and horizontal plane) of the building and load conditions play a key role [7,8]. Over time, builders in earthquake-prone areas have implemented empirical strengthening techniques by trial and error to enhance the performance of URM structures, such as ties, buttresses, and timber skeletons [10]. However, it is in recent decades that the scientific community has begun to systematically study these techniques with a scientific approach. Additionally, advancements in technology have enabled researchers and builders to develop methods using contemporary materials. A brief review of the main strengthening techniques is presented in Section 2. The accurate prediction and assessment of the damage evolution in existing URM buildings under seismic actions remain challenging tasks due to the complexity of the material and the interaction between all the structural elements, along with the nature of the earthquakes [11]. Researchers have developed a wide variety of mathematical models and strategies for this type of structure, aiming to provide more efficient procedures while ensuring reliable results [12]. The selection of mechanical properties depends on the chosen modelling strategy and applied analysis method [13]. Section 3 presents an overview of different modelling strategies and types of analyses, as well as time integration methods used to solve the system of equations, i.e., implicit and explicit solvers. For the sake of brevity, the most important features of these methods are highlighted, while mathematical formulations are not described. Figure 1: IP failure modes (adapted from [9]): (**a**) flexural failure; (**b**) shear-diagonal failure; (**c**) shear-sliding failure.