This brief focuses on the structural properties of nonlinear time-delay systems. It provides a link between coverage of fundamental theoretical properties and advanced control algorithms, as well as suggesting a path for the generalization of the differential geometric approach to time-delay systems . The brief begins with an introduction to a class of single-input nonlinear time-delay systems. It then focuses on geometric methods treating them and offers a geometric framework for integrability. The book has chapters dedicated to the accessibility and observability of nonlinear time-delay systems, allowing readers to understand the systems in a well-ordered, structured way. Finally, the brief concludes with applications of integrability and the control of single-input time-delay systems. This brief employs exercises and examples to familiarize readers with the time-delay context. It is of interest to researchers, engineers and postgraduate students who work in the area of nonlinear control systems.