Model comparison has been widely used to support many tasks in model-driven software development. For this reason, many techniques of comparing them have been proposed in the last few decades. However, academia and industry have overlooked a classi cation of currently available approaches to the comparison of design models. Hence, a thorough understanding of state-of-the-art techniques remains limited and incon- clusive. This article, therefore, focuses on providing a classi cation and a thematic analysis of studies on the comparison of software design models. We carried out a systematic mapping study following well-established guidelines to answer nine research questions. In total, 56 primary studies (out of 4,132) were selected from 10 widely recognized electronic databases after a careful ltering process. The main results are that a majority of the primary studies (1) provide coarse-grained techniques of the comparison of general-purpose diagrams, (2) adopt graphs as principal data structure and compare software design models considering structural prop- erties only, (3) pinpoint commonalities and di erences between software design models rather than assess their similarity, and (4) propose new techniques while neglecting the production of empirical knowledge from experimental studies. Finally, this article highlights some challenges and directions that can be explored in upcoming studies.