Dynamic and static behaviors of a three-dimensional catenary system for a high-speed railway are analyzed by using the finite element method. Considering tensions in the contact wire and the messenger wire, we drive the equations of motion for the catenary system. These equations are for the longitudinal, transverse, vertical and torsional motions. After establishing the weak form, the weak forms are spatially discretized with newly defined two-node beam elements. With the discretized equations, a finite element computer program is developed for the static and dynamic analyses. The static deflections of the catenary system, which are important for good contact between the pantograph and the contact line, are computed when the gravity is applied. On the other hand, we analyze the natural frequencies and the corresponding natural modes of the catenary system. The dynamic responses of the system are also investigated when applying a load to the contact line. For verification of the developed finite element program, vibrations of the catenary system are measured and they are compared to computed time responses.