Large Language Models (LLMs) have demonstrated significant potential in medicine, with many studies adapting them through continued pre-training or fine-tuning on medical data to enhance domain-specific accuracy and safety. However, a key open question remains: to what extent do LLMs memorize medical training data. Memorization can be beneficial when it enables LLMs to retain valuable medical knowledge during domain adaptation. Yet, it also raises concerns. LLMs may inadvertently reproduce sensitive clinical content (e.g., patient-specific details), and excessive memorization may reduce model generalizability, increasing risks of misdiagnosis and making unwarranted recommendations. These risks are further amplified by the generative nature of LLMs, which can not only surface memorized content but also produce overconfident, misleading outputs that may hinder clinical adoption. In this work, we present a study on memorization of LLMs in medicine, assessing its prevalence (how frequently it occurs), characteristics (what is memorized), volume (how much content is memorized), and potential downstream impacts (how memorization may affect medical applications). We systematically analyze common adaptation scenarios: (1) continued pretraining on medical corpora, (2) fine-tuning on standard medical benchmarks, and (3) fine-tuning on real-world clinical data, including over 13,000 unique inpatient records from Yale New Haven Health System. The results demonstrate that memorization is prevalent across all adaptation scenarios and significantly higher than that reported in the general domain. Moreover, memorization has distinct characteristics during continued pre-training and fine-tuning, and it is persistent: up to 87% of content memorized during continued pre-training remains after fine-tuning on new medical tasks.