Masonry buildings collect a wide variety of constructions, which are characterized by very different types of masonry and structural systems, moving through historical periods and geographical areas. Within this context, in the chapter, after a brief introduction on the seismic behaviour of masonry structures (§5.2) and the critical issues that affect existing buildings (§5.3), specific criteria for the application of the Displacement Based Assessment (DBA) are illustrated with reference to criteria for definition of: limit states (§5.5); equivalent damping (§5.6); and recurring deformed shapes (§5.7). Tools applicable both for a preliminary assessment of the existing buildings (suitable also for assessment at large scale) and the final one are proposed and discussed. The modelling tools adopted and the cases studied examined are illustrated in §5.4; these latters have been selected to be representative of some of the relevant features of existing masonry buildings, like the presence of weak spandrels, flexible floors and irregularities. Among masonry types aforementioned, the attention is focused on the response of Unreinforced Masonry (URM) buildings, that constitute the majority of existing stock in Italy, and on the global response. Despite this, the criteria proposed (e.g. the multi-scale approach for the definition of limit states proposed in §5.5) have general validity and are applicable also to other types. Moreover, results on the specific behaviour and possible vulnerability of other classes, such as the mixed ones, are illustrated in previous research done by the UNIGE group. Finally, in §5.8 some brief remarks on the probabilistic seismic assessment are illustrated; in particular, this topic has been deepened within the research done in collaboration with Prof. P.E.Pinto and P.Franchin in the RS4 Special Project of Reluis.