In this paper we study the Cauchy problem for some classes of weakly coupled systems of semilinear damped σ-evolution equations with unbounded coefficients of the form { u tt + a ( x )(– ∆ ) σ u + u t = F 1 ( v , v t ), v tt + a ( x )(– ∆ ) σ u + u t = F 2 ( u , u t ), u ( 0 , x )= u 0 , u t ( 0 , x )= u 1 , v ( 0 , x )= v 0 , v t ( 0 , x )= v 1 , where σ∈(0,1), a( x) is a continuous function satisfying a ( x ) ≈ | x | α a.e. in R n with α∈[0,2 σ), the nonlinearities f ( v , v t ) and g ( u , u t ) are assumed to be of the form | v | p or | v t | p , and | u | q or | u t | q , respectively. Our main aim is to prove the global (in time) existence and decay rates of mild solutions to these systems with small initial data belonging to the weighted space L ρ 2 ∩ L ρ m , where ρ : = ρ ( x ) = a ( x ) – 1 . The proof is based on suitable linear estimates and the fixed point method using the Banach contraction mapping principle. A blow-up result in the case of systems with power nonlinearities is also given to illustrate the optimality of the corresponding global existence result.