General second order quasilinear elliptic systems with nonlinear boundary conditions on bounded domains are formulated into nonlinear mappings between Sobolev spaces. It is shown that the linearized mapping is a Fredholm operator of index zero. This and the abstract global bifurcation theorem of [Jacobo Pejsachowicz, Patrick J. Rabier, Degree theory for C1 Fredholm mappings of index 0, J. Anal. Math. 76 (1998) 289–319] allow us to carry out bifurcation analysis directly on these elliptic systems. At the abstract level, we establish a unilateral global bifurcation result that is needed when studying positive solutions. Finally, we supply two examples of cross-diffusion population model and chemotaxis model to demonstrate how the theory can be applied.