A structure is vulnerable if any damage produces consequences which are disproportionate to that damage. Conversely a structure is robust if it can withstand arbitrary damage. The theory of vulnerability developed in this thesis concerns structural form and connectivity. The purpose of the theory is to identify the "weak links" within a structure. A new theory of vulnerability for 3D structures has been developed. An entirely new approach has been derived from the 2D theory developed by Wu, Blockley, Woodman (1993), Yu(1997). New concepts of a graph model for 3D structural systems, 3D structural path and loop, structural round, string pattern of structural round, 3D structural tightness, 3D structural clusters, deterioration hierarchy of structural round etc. have been defined and introduced. A method of combining structural vulnerability analysis with structural response analysis has been developed and illustrated with three examples. 2D structural vulnerability theory as developed most recently by Yu(1997) has also been re-examined and improved. The algorithm for computer implementation of 2D structural vulnerability theory has been greatly improved. The improved algorithm provides a more complete search, always use the most up to date hierarchy and has a special treatment for structure with a mixture of types of joint. In order to create a graphical user interface for SAVE (Structural Analysis for Vulnerability Estimation) (Yu, 1997), some graphical functions have been developed. These graphical functions include: drawing the structure; drawing the cluster formation procedures; drawing the hierarchical presentation; drawing the most vulnerable failure scenarios in the structure. The theory is potentially of use in the design of protection and performance monitoring programmes for safety management and to reduce the risk of overlooking vulnerable failure scenarios