An Introductory Geometrical Application: 1.1 Nested triangles; 1.2 The transformation $\sigma$; 1.3 The transformation $\sigma$, iterated with different values of $s$; 1.4 Nested polygons Introductory Matrix Material: 2.1 Block operations; 2.2 Direct sums; 2.3 Kronecker product; 2.4 Permutation matrices; 2.5 The Fourier matrix; 2.6 Hadamard matrices; 2.7 Trace; 2.8 Generalized inverse; 2.9 Normal matrices, quadratic forms, and field of values Circulant Matrices: 3.1 Introductory properties; 3.2 Diagonalization of circulants; 3.3 Multiplication and inversion of circulants; 3.4 Additional properties of circulants; 3.5 Circulant transforms; 3.6 Convergence questions Some Geometric Applications of Circulants: 4.1 Circulant quadratic forms arising in geometry; 4.2 The isoperimetric inequality for isosceles polygons; 4.3 Quadratic forms under side conditions; 4.4 Nested $n$-gons; 4.5 Smoothing and variation reduction; 4.6 Applications to elementary plane geometry: $n$-gons and $K_r$-grams; 4.7 The special case: $\text{circ}(s, t, 0, 0, \dots, 0)$; 4.8 Elementary geometry and the Moore-Penrose inverse Generalizations of Circulants: $g$-Circulants and Block Circulants: 5.1 $g$-circulants; 5.2 $0$-circulants; 5.3 PD-matrices; 5.4 An equivalence relation on $\{1, 2, \dots, n\}$; 5.5 Jordanization of $g$-circulants; 5.6 Block circulants; 5.7 Matrices with circulant blocks; 5.8 Block circulants with circulant blocks; 5.9 Further generalizations Centralizers and Circulants; 6.1 The leitmotiv; 6.2 Systems of linear matrix equations. The centralizer; 6.3 $\div$ algebras; 6.4 Some classes $Z(P_{\sigma}, P_{\tau})$; 6.5 Circulants and their generalizations; 6.6 The centralizer of $J$; magic squares; 6.7 Kronecker products of $I, \pi$, and $J$; 6.8 Best approximation by elements of centralizers Appendix Bibliography Index of authors Index of subjects