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We study expressions of the form \\[\n  \\sum_{h \\bmod{a_k}} \\ \\prod_{\\substack{i=1\\\\ i\\not=k}}^{n} \\ \\B_{p_i}\\left(a_i \\frac{h+x_k}{a_k}-x_i\\right). \\] These \\highlight{Bernoulli--Dedekind sums} generalize and unify various arithmetic sums introduced by Dedekind, Apostol, Carlitz, Rademacher, Sczech, Hall--Wilson--Zagier, and others. 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