The process of carrying out similarity and analogy comparisons can be modeled as the alignment and mapping of structured representations. In this paper, we focus on three aspects of comparison that are central in structure-mapping theory. All three have occasioned controversy. First, comparison involves structured representations. Second, the comparison process is driven by a preference for connected relational structure. Third, the mapping between domains is rooted in semantic similarity between the relations that characterize the domains. For each of these points, we review supporting evidence and discuss some challenges that have been raised by other researchers. We end with a discussion of the role of structure-mapping in other cognitive processes.