University students' success in core mathematics courses has been linked to their overall academic achievement in many undergraduate programs. This finding has prompted an ongoing inquiry into the way mathematics is taught, focusing, specifically, on the university mathematics lecture--the most common instructional activity--and on the extent to which it is conducive to student engagement, regarded as key to successful learning. This dissertation reports on a multimodal study of student engagement in undergraduate mathematics lectures with the purpose to develop a context-specific understanding of student engagement, and identify and describe aspects of university mathematics teaching that impact student engagement. A theoretical and analytical framework combining concepts from discourse studies, Vygotskian sociocultural theory, situated learning, multimodality, speech theory, and gesture studies is used to investigate university mathematics teaching as a complex discursive practice. The study responds to the following research questions: 1) What constitutes student engagement in the university mathematics teaching context? How can it be defined? 2) What aspects of the classroom context affect student engagement? and 3) What strategies do university mathematics instructors use to help realize student engagement in lectures? A mixed methods study with an explanatory sequential design was conducted wherein the results of the first phase of data collection and analysis, involving student surveys and interviews with university mathematics instructor participants, informed the second phase involving the collection and analysis of video-recorded lectures, observational field notes, and follow-up interviews with instructors. Based on the study findings, a context-specific definition of student engagement in mathematics lectures is developed, and the aspects of the learning environment and classroom climate impacting student engagement are identified and discussed. The study indicates that by using embodied multimodal patterned strategies, university mathematics instructors facilitate student engagement by establishing and maintaining a continuous focused interaction with the students and by involving students in the collaborative process of doing mathematics in real-time. The study outcomes have implications for future research into university lecturing in face-to-face and online teaching contexts, and the development of programs and materials for university teaching and training new faculty.