We present a review of recent techniques for performing geometric analysis in cultural heritage (CH) applications. The survey is aimed at researchers in the areas of computer graphics, computer vision and CH computing, as well as to scholars and practitioners in the CH field. The problems considered include shape perception enhancement, restoration and preservation support, monitoring over time, object interpretation and collection analysis. All of these problems typically rely on an understanding of the structure of the shapes in question at both a local and global level. In this survey, we discuss the different problem forms and review the main solution methods, aided by classification criteria based on the geometric scale at which the analysis is performed and the cardinality of the relationships among object parts exploited during the analysis. We finalize the report by discussing open problems and future perspectives.