Using the PRISMA approach, we present the first systematic literature review of digital twin (DT) research in healthcare systems (HSs). This endeavor stems from the pressing need for a thorough analysis of this emerging yet fragmented research area, with the goal of consolidating knowledge to catalyze its growth. Our findings are structured around three research questions aimed at identifying: (i) current research trends, (ii) gaps, and (iii) realization challenges. Current trends indicate global interest and interdisciplinary collaborations to address complex HS challenges. However, existing research predominantly focuses on conceptualization; research on integration, verification, and implementation is nascent. Additionally, we document that a substantial body of papers mislabel their work, often disregarding modeling and twinning methods that are necessary elements of a DT. Furthermore, we provide a non-exhaustive classification of the literature based on two axes: the object (i.e., product or process) and the context (i.e., patient's body, medical procedures, healthcare facilities, and public health). While this is a testament to the diversity of the field, it implies a specific pattern that could be reimagined. We also identify two gaps: (i) considering the human-in-the-loop nature of HSs with a focus on provider decision-making and (ii) implementation research. Lastly, we discuss two challenges for broad-scale implementation of DTs in HSs: improving virtual-to-physical connectivity and data-related issues. In conclusion, this study suggests that DT research could potentially help alleviate the acute shortcomings of HSs that are often manifested in the inability to concurrently improve the quality of care, provider wellbeing, and cost efficiency.