This guide is a compilation of lectures that I have refined for publication, which I have delivered to mathematics students over many years. I began teaching the theory of determinants with an introductory course for first-semester students. Later, I revisited this topic in my advanced algebra course, and while teaching the integration of differential equations, I provided a thorough exposition on functional determinants. As a result, students received a comprehensive understanding of this discipline throughout the course, which has found applications in every branch of mathematics today. Acknowledging the significance of determinants, I aimed to teach in a way that students would grasp both the methodology for developing this theory and the various techniques for calculating determinants. In this guide, I have supplemented my lectures with the most fundamental applications of determinants to the algebraic theory of forms, as well as a discussion on determinants of infinite order. I place particular emphasis on this latter section due to the pivotal role that infinite order determinants have played in connection with the work of Fredholm; they have become a cornerstone of a new branch of analysis that his research has led to. Furthermore, to my knowledge, this section of the theory of determinants is presented in detail in Russian literature for the first time.