Let G be a finite and simple graph with vertex set V ( G ) , and let f : V ( G ) → { − 1 , 1 } be a two-valued function. If ∑ x ∈ N [ v ] f ( x ) ≥ 1 for each v ∈ V ( G ) , where N [ v ] is the closed neighborhood of v , then f is a signed dominating function on G . A set { f 1 , f 2 , … , f d } of signed dominating functions on G with the property that ∑ i = 1 d f i ( x ) ≤ 1 for each x ∈ V ( G ) , is called a signed dominating family (of functions) on G . The maximum number of functions in a signed dominating family on G is the signed domatic number on G . In this paper, we investigate the signed domatic number of some circulant graphs and of the torus C p × C q .