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In this paper, we define almost $(m,n) $-ideals of semigroups by using the concepts of $(m,n) $-ideals and almost ideals of semigroups. An almost $(m,n) $-ideal is a generalization of $(m,n) $-ideals and a generalization of almost one-sided ideals. We investigate properties of almost $(m,n) $-ideals of semigroups. Moreover, we define fuzzy almost $(m,n) $-ideals of semigroups and give relationships between almost $(m,n) $-ideals and fuzzy almost $(m,n) $-ideals.