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We introduce a notion of Gieseker stability for coherent sheaves on tameDeligne-Mumford stacks with projective moduli scheme and some chosen generatingsheaf on the stack in the sense of Olsson and Starr \cite{MR2007396}. We provethat this stability condition is open, and pure dimensional semistable sheavesform a bounded family. We explicitly construct the moduli stack of semistablesheaves as a finite type global quotient, and study the moduli scheme of stablesheaves and its natural compactification in the same spirit as the seminalpaper of Simpson \cite{MR1307297}. With this general machinery we are able toretrieve, as special cases, results of Lieblich \cite{MR2309155} and Yoshioka\cite{MR2306170} about moduli of twisted sheaves and parabolic stabilityintroduced by Maruyama-Yokogawa in \cite{MR1162674}.

Moduli Spaces of Semistable Sheaves on Projective Deligne-Mumford Stacks