Background: The wellbeing of college students today is at risk because of rising occurrences of mental health issues in higher education. Concurrently, undergraduate students perceive engineering courses and programs to be among the most arduous, and least welcoming and accommodating, in higher education. Although research related to mental health and wellbeing (MHW) in engineering is growing, a systematic review of this research has yet to be conducted. Purpose: This systematic literature review identifies and synthesizes empirical scholarship related to the MHW of undergraduate engineering students. Scope/Method: Specified search terms and inclusion criteria were used to identify 34 empirical studies related to engineering undergraduates' MHW. Content and qualitative thematic analyses were conducted to characterize and synthesize trends in research quality and outcomes across studies. Results: Undergraduate engineering students experience a variety of mental health issues that negatively affect their experiences in engineering education. Stress is the most prevalent mental health issue identified; anxiety, depression, and post-traumatic stress disorder (PTSD) are also reported. Heavy academic workloads, sleep issues, and the nature of engineering education culture are identified as impediments to MHW in engineering education. Conclusions: Although MHW in engineering is a growing area of research internationally, current MHW research in engineering is nascent and focused on the characterization of student mental health issues. Researchers underutilize qualitative and mixed-methods approaches, longitudinal and experimental designs, guiding frameworks, and robust sampling techniques. Academic and mindfulness interventions, as well as the use of mental wellness constructs from positive psychology, show promise for supporting MHW in engineering.