Understanding a complex system of relationships between courses is of great importance for the university's educational mission. This paper is dedicated to the study of course-prerequisite networks (CPNs), where nodes represent courses and directed links represent the formal prerequisite relationships between them. The main goal of CPNs is to model interactions between courses, represent the flow of knowledge in academic curricula, and serve as a key tool for visualizing, analyzing, and optimizing complex curricula. First, we consider several classical centrality measures, discuss their meaning in the context of CPNs, and use them for the identification of important courses. Next, we describe the hierarchical structure of a CPN using the topological stratification of the network. Finally, we perform the interdependence analysis, which allows to quantify the strength of knowledge flow between university divisions and helps to identify the most intradependent, influential, and interdisciplinary areas of study. We discuss how course-prerequisite networks can be used by students, faculty, and administrators for detecting important courses, improving existing and creating new courses, navigating complex curricula, allocating teaching resources, increasing interdisciplinary interactions between departments, revamping curricula, and enhancing the overall students' learning experience. The proposed methodology can be used for the analysis of any CPN, and it is illustrated with a network of courses taught at the California Institute of Technology. The network data analyzed in this paper is publicly available in the GitHub repository.