Some mathematical topics, circulant matrices, in particular, are pure gems that cry out to be admired and studied with different techniques or perspectives in mind. Our work on this subject was originally motivated by the apparent need of one of the authors (IK) to derive a specific result, in the spirit of Proposition 24, to be applied in his investigation of theta constant identities [9]. Although progress on that front eliminated the need for such a theorem, the search for it continued and was stimulated by enlightening conversations with Yum-Tong Siu during a visit to Vietnam. Upon IK’s return to the US, a visit by Paul Fuhrmann brought to his attention a vast literature on the subject, including the monograph [4]. Conversations in the Stony Brook Mathematics’ common room attracted the attention of the other author, and that of Sorin Popescu and Daryl Geller∗ to the subject, and made it apparent that circulant matrices are worth studying in their own right, in part because of the rich literature on the subject connecting it to diverse parts of mathematics. These productive interchanges between the participants resulted in [5], the basis for this article. After that version of the paper lay dormant for a number of years, the authors’ interest was rekindled by the casual discovery by SRS that these matrices are connected with algebraic geometry over the mythical field of one element. Circulant matrices are prevalent in many parts of mathematics (see, for example, [8]). We point the reader to the elegant treatment given in [4, §5.2], and to the monograph [1] devoted to the subject. These matrices appear naturally in areas of mathematics where the roots of unity play a role, and some of the reasons for this to be so will unfurl in our presentation. However ubiquitous they are, many facts about these matrices can be proven using only basic linear algebra. This makes the area quite accessible to undergraduates looking for research problems, or mathematics teachers searching for topics of unique interest to present to their students. We concentrate on the discussion of necessary and sufficient conditions for circulant matrices to be non-singular, and on various distinct representations they have, goals that allow us to lay out the rich mathematical structure that surrounds them. Our treatement though is by no means exhaustive. We expand on their connection to the algebraic geometry over a field with one element, to normal curves, and to Toeplitz’s operators. The latter material illustrates the strong presence these matrices have in various parts of modern and classical mathematics. Additional connections to other mathematics may be found in [11]. The paper is organized as follows. In §2 we introduce the basic definitions, and present two models of the space of circulant matrices, including that as a