A research object of this study is a extrusion of the composite rods. As the condition of the incompressibility, a set of stream functions was proposed by this study to construct analysis model of the conventional and the hydrostatic extrusion. The upper-bound approach was applied to investigate the plastic deformation behavior during composite rods extrusion. The topical subject of this study was consisted of conventional extrusion, hydrostatic extrusion and the instability of composite materials during conventional extrusion. This study proposes a set of stream functions to investigate the plastic deformation behavior of the rods during axisymmetric extrusion of composite rods through a conical die. The velocity field and the rigid-plastic boundaries can be obtained from the proposed stream functions immediately. Relative slip at the interface of the rods is allowed. The friction factor between the metal and die is determined by a ring compression test. By using the upper-bound approach, the radius ratio of the core layer at the exit of the die and the plastic region within the die are determined by the minimization of the total power. Experiments on extrusion of composite rods composed of aluminum (core layer) and copper (sleeve layer) are also carried out. The predicted force, final radius ratio and flow pattern of the rods within the die are compared with the experimental results. Being different with conventional extrusion, a condition that has relative slip at the interface of the rods at the exit of the die was modified to become no relative slip. The same set of stream functions with that of the conventional extrusion was also applied to simulate the flow behavior of the composite rods through a conical die during hydrostatic extrusion. This model is used to predict the hydrostatic extrusion pressure for various process parameters via the upper bound theorem. Experiments are also carried out using composite rods composed of copper as the sleeve layer and commercial aluminum as the core layer. The experimental results are compared with the theoretical calculations. A plastic instability criterion under a tri-axial stress state was developed in this study to predict the occurrence of plastic instability in the hard core layer during axisymmetric composite rod extrusion. Stress states in the hard core layer at the exit of the die, where plastic instability is most likely to occur, were employed to formulate the plastic instability criterion. The effects of various extrusion conditions, such as the initial radius ratio and yield stress ratio of the composite rods, etc., upon the critical area reduction, under which plastic instability can be avoided, were discussed systematically. Furthermore, experiments on composite rod extrusion were also conducted by employing aluminum and lead as layers of composite rods. It was found that the theoretical predictions of the critical area reduction were in good agreement with the experimental measurements with the initial area of 0.15.