The construction and properties of group matrices are analyzed in more detail. The group matrix of a cyclic group with a certain ordering is a circulant matrix. The book by Davis (Circulant Matrices, Chelsea, New York, 1994) gives a comprehensive account of circulants and the chapter is designed to provide a far reaching extension and generalization of the results there. If an arbitrary subgroup H of a group G is taken, it is shown that with an appropriate ordering the group matrix XG is a block matrix in which each block is of the form where the elements in are all distinct and of the form This leads to methods to partially diagonalize the group matrix which facilitate the calculation of the group determinant. Group matrix versions of the ring of representations and the Burnside ring of a group are described using supermatrices. A description of projective group matrices, corresponding to projective representations, is given. Work of L. E. Dickson on group matrices and group determinants over a finite field is also described.