The authors initially planned to write an article describing the origins and devel- opments of the theory of Fredholm operators and to present their recollections of this topic. We started to read again classical papers and we were sidetracked by the literature concerned with the theory and applications of traces and determi- nants of infinite matrices and integral operators. We were especially impressed by the papers of Poincare, von Koch, Fredholm, Hilbert and Carleman, as well as F. Riesz's book on infinite systems of linear equations. Consequently our plans were changed and we decided to write a paper on the history of determinants of infi- nite matrices and operators. During the preparation of our paper we realized that many mathematical questions had to be answered in order to gain a more com- plete understanding of the subject. So, we changed our plans again and decided to present the subject in a more advanced form which would satisfy our new require- ments. This whole process took between four and five years of challenging, but enjoyable work. This entailed the study of the appropriate relatively recent results of Grothendieck, Ruston, Pietsch, Hermann Konig and others. After the papers [GGK1] and [GGK2] were published, we saw that the written material could serve as the basis of a book.