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Let $D$ be a combinatorial design and denote by $\\gamma(D)$ the domination number of the incidence (Levy) graph of $D$. We obtain a number of results about the domination numbers of various kinds of designs.\n  For instance, a finite projective plane of order $n$, which is a symmetric $(n^{2}+n+1,n+1,1)$-design, has $\\gamma=2n$. %We also show that for any symmetric $(v,k,\\lambda)$-design it holds that $\\gamma \\leq 2k$. 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