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In 2002, Grieves defined the concept of the digital twin as a virtual instance of physical assets capable of continuously mirroring them. Ever since then, driven by remarkable industrial attention, digital twins flourished and ripened in several sectors. The notable industrial adoption has been sided by a growing interest from the software engineering community in general and the software architecture community in particular as demonstrated by the growing number of published peer-reviewed publications and proposed software architectural solutions for digital twins. In this paper, we report on the planning, execution, and results of a systematic mapping study on architecting digital twins. The study captures crucial aspects of software architectures for digital twins as types of architectural solutions, quality attributes, and architectural patterns. It supports practitioners in creating digital twins tailored to their specific needs and researchers in identifying trends and open challenges. Starting from an initial set of potentially relevant 1630 peer-reviewed publications, we selected 140 primary studies. We analysed the set of primary studies using thorough data extraction, analysis, and synthesis process. To compensate for single method limitations and reduce possible threats to conclusion validity, we discussed the results of our study with experts in the software architecture community. Based on our results, the field of software architecture for digital twins is lively and an increasing number of architectural solutions are being proposed. Although there is a lack of widely accepted reference architectural solutions for digital twins, most of them are built using a combination of the layered and service-oriented patterns and address maintainability, performance efficiency, and compatibility quality attributes.