Many universities have recently focused significant efforts on enhancing their graduation rates. Numerous factors may impact a student's ability to succeed and ultimately graduate, including pre-university preparation, as well as the student support services provided by a university. However, even the best efforts to improve in these areas may fail if other institutional factors overwhelm their ability to facilitate student progress. Specifically, in this paper we consider degree to which the underlying curriculum that a student must traverse in order to earn a degree impacts progress. Using complex network analysis and graph theory, this paper proposes a framework for analyzing university course networks at the university, college and departmental levels. The analyses we provide are based on quantifying the importance of a course based on its delay and blocking factors, as well as the number of curricula that incorporate the course, leading to a metric we refer to as the course cruciality. Experimental results, using data from the University of New Mexico, show that the distribution of course cruciality follows a power law distribution. Applications of the proposed framework are extended to study the complexity of curricula within colleges as well as the tendency of a university's disciplines to associate with others that are unlike them. This work may be useful to both students and decision makers at universities as it presents a robust framework for analyzing the ease of flow of students through curricula, which may lead to improvements that facilitate improved student success.