  The matrix sine-Gordon theory, a matrix generalization of the well-known
sine-Gordon theory, is studied. In particular, the $A_{3}$-generalization where
fields take value in $SU(2)$ describes integrable deformations of conformal
field theory corresponding to the coset $SU(2) \times SU(2) /SU(2)$. Various
classical aspects of the matrix sine-Gordon theory are addressed. We find exact
solutions, solitons and breathers which generalize those of the sine-Gordon
theory with internal degrees of freedom, by applying the Zakharov-Shabat
dressing method and explain their physical properties. Infinite current
conservation laws and the B\"{a}cklund transformation of the theory are
obtained from the zero curvature formalism of the equation of motion. From the
B\"{a}cklund transformation, we also derive exact solutions as well as a
nonlinear superposition principle by making use of the Bianchi's permutability
theorem.
