Proper modelling of a dynamic system can benefit analysis, simulation, design, evaluation and control of the system. The linear-graph (LG) approach is suitable for modelling lumped-parameter dynamic systems. By using the concepts of graph trees, it provides a graphical representation of the system, with a direct correspondence to the physical component topology. This paper systematically extends the application of LGs to multi-domain (mixed-domain or multi-physics) dynamic systems by presenting a unified way to represent different domains - mechanical, electrical, thermal and fluid. Preservation of the structural correspondence across domains is a particular advantage of LGs when modelling mixed-domain systems. The generalisation of Thevenin and Norton equivalent circuits to mixed-domain systems, using LGs, is presented. The structure of an LG model may follow a specific pattern. Vector LGs are introduced to take advantage of such patterns, giving a general LG representation for them. Through these vector LGs, the model representation becomes simpler and rather compact, both topologically and parametrically. A new single LG element is defined to facilitate the modelling of distributed-parameter (DP) systems. Examples are presented using multi-domain systems (a motion-control system and a flow-controlled pump), a multi-body mechanical system (robot manipulator) and DP systems (structural rods) to illustrate the application and advantages of the methodologies developed in the paper.