The complex phase diagram of high-critical temperature (T-c) superconductors can be deduced from an SO(5) symmetry principle that unifies antiferromagnetism and d-wave superconductivity. The approximate SO(5) symmetry has been derived from the microscopic Hamiltonian, and it becomes exact under renormalization group flow toward a bicritical point. This symmetry enables the construction of a SO(5) quantum nonlinear sigma model that describes the phase diagram and the effective low-energy dynamics of the system. This model naturally explains the basic phenomenology of the high-T-c superconductors from the insulating to the underdoped and the optimally doped region.