We consider a class of weakly coupled systems of elliptic operators A with unbounded coefficients defined in R N . We prove that a semigroup (T (t)) t≥0 of bounded linear operators can be associated with A, in a natural way, in the space of all bounded and continuous functions. We prove a compactness property of the semigroup as well as some uniform estimates on the derivatives of the function T (t)f , when f belongs to some spaces of Hölder continuous functions, which are the key tools to prove some optimal Schauder estimates for the solution to some nonhomogeneous elliptic equations and Cauchy problems associated with the operator A. Under suitable additional conditions, we then prove that the restriction of the semigroup to the subspace of smooth and compactly supported functions extends by a strongly continuous semigroup to L p -spaces over R N , related to the Lebesgue measure, when p ∈ [1, ∞). We also provide sufficient conditions for this semigroup to be analytic when p ∈ (1, ∞). Finally, we prove some L p -L q -estimates.Mathematics Subject Classification (2010). Primary 47D06; Secondary 35B45, 35B65, 35K40, 60J35.