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Abstract. We construct algebraic moduli stacks of log structures and give stack-theoretic interpretations of K. Kato's notions of log flat, log smooth, and logétale morphisms. In the last section we describe the local structure of these moduli stacks in terms of toric stacks.

In this paper we develop a theory of Grothendieck's six operations of lisse-étale constructible sheaves on Artin stacks locally of finite type over certain excellent schemes of finite Krull dimension. We also give generalizations of the classical base change theorems and Kunneth formula to stacks, and prove new results about cohomological descent for unbounded complexes.

Abstract. Given a separated and locally finitely-presented DeligneMumford stack X over an algebraic space S, and a locally finitelypresented O X -module F , we prove that the Quot functor Quot(F /X /S) is represented by a separated and locally finitely-presented algebraic space over S. Under additional hypotheses, we prove that the connected components of Quot(F /X /S) are quasi-projective over S.

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Abstract. In this paper we develop a theory of Grothendieck's six operations for adic constructible sheaves on Artin stacks continuing the study of the finite coefficients case in [12].

We describe an equivalence between the notion of balanced twisted curve introduced by Abramovich and Vistoli, and a new notion of log twisted curve, which is a nodal curve equipped with some logarithmic data in the sense of Fontaine and Illusie. As applications of this equivalence, we construct a universal balanced twisted curve, prove that a balanced twisted curve over a general base scheme admitsétale locally on the base a finite flat cover by a scheme, and also give a new construction of the moduli space of stable maps into a Deligne-Mumford stack and a new proof that it is bounded.