  In this paper, from the algebraic reductions from the Lie algebra
$gl(n,\mathbb C)$ to its commutative subalgebra $Z_n$, we construct the general
$Z_n$-Sine-Gordon and $Z_n$-Sinh-Gordon systems which contain many
multi-component Sine-Gordon type and Sinh-Gordon type equations. Meanwhile, we
give the B\"acklund transformations of the $Z_n$-Sine-Gordon and
$Z_n$-Sinh-Gordon equations which can generate new solutions from seed
solutions. To see the $Z_n$-systems clearly, we consider the $Z_2$-Sine-Gordon
and $Z_3$-Sine-Gordon equations explicitly including their B\"acklund
transformations, the nonlinear superposition formula and Lax pairs.
