Determinants declined in prestige from the mid-nineteenth century onwards and are now best known for their applications in matrix theory, where they appear in a subsidiary role. However, during the last thirty years determinants have arisen independently of matrices in the solution of problems arising in the reflectionless transmission of electromagnetic waves through dielectrics, the Korteweg-de Vries equation, relativistic gravitational fields, and other branches of theoretical physics. Teoplitz determinants, which arise in statistical mechanics, have interesting properties in their own right and are no mere appendages to their parent matrices. The object of this paper is to survey very briefly these and other applications of determinants and to note some recent developments in determinant theory. There are 56 references.