Geometric modeling is the main research area in computer-aided design (CAD), computer-aided engineering (CAE), and computer-aided manufacturing (CAM), focusing on the representation and design of geometric models on computers. Currently, the standard representation is non-uniform rational B-splines (NURBS). However, due to the tensor-product structure, NURBS has several main limitations in terms of the representations of complex models. T-spline is a new NURBS compatible free-form surface representation technology. T-splines have drawn wide attention both from academia and industry, after solving several limitations of the standard NURBS representation. To enable a more comprehensive understanding of T-splines, a systematic review was conducted. Based on the extensive literature research, the definition and basic operations for T-splines and applications of T-splines in CAD, CAE and CAM were summarized and compared, which mainly focused on the basic ideas, theories, merits and demerits of various algorithms. Due to the high demand of accuracy and efficiency required by geometric representation in the industry, the research on T-spline remains imperfect, and a number of problems and possible prospects are in need of exploration.