Large Language Models (LLMs) have demonstrated significant potential in medicine. To date, LLMs have been widely applied to tasks such as diagnostic assistance, medical question answering, and clinical information synthesis. However, a key open question remains: to what extent do LLMs memorize medical training data. In this study, we present the first comprehensive evaluation of 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 reported in the general domain. Memorization affects both the development and adoption of LLMs in medicine and can be categorized into three types: beneficial (e.g., accurate recall of clinical guidelines and biomedical references), uninformative (e.g., repeated disclaimers or templated medical document language), and harmful (e.g., regeneration of dataset-specific or sensitive clinical content). Based on these findings, we offer practical recommendations to facilitate beneficial memorization that enhances domain-specific reasoning and factual accuracy, minimize uninformative memorization to promote deeper learning beyond surface-level patterns, and mitigate harmful memorization to prevent the leakage of sensitive or identifiable patient information.