In this article, the orbital angular momentum (OAM) mode system, known from antenna communication, is applied and extended to various multiconductor transmission lines. In the context of this work, an OAM mode describes the excitation of a multiport system with a voltage distribution that is an eigenmode of a cyclic system. The OAM mode system is the complete system composed of all $N$ OAM modes for a $N$ -port system. For two conductors, the OAM mode system is shown to be equivalent to the differential- and common-mode systems. A transmission line carrying OAM modes must be used as a feed line in OAM communication links and, because of the orthogonality of the modes, it can be used to transmit multiple signals without mode conversion. An OAM communication link is defined in the context of this work as the components used from mode excitation of the transmitter to mode decomposition of the receiver. Using the analogy of OAM communication with antenna arrays and transmission-line theory, the excitation, geometry, characteristic mode impedance, mode power, and matching of a transmission line carrying OAM modes are derived. The matching is implemented with a mode decomposition and mode matching network on both sides of the transmission line. The performance of the system is verified numerically, with a 2-D boundary element method (BEM) solver based on the method of moments. Implementations of the OAM mode system for multiconductor cables, microstrip lines, and striplines are examined. In addition, the behavior of the mode transmission if the OAM modes are mismatched is evaluated and compared to a single conductor transmission line.