  In this paper we study the homotopy limits of cosimplicial diagrams of
dg-categories. We first give an explicit construction of the totalization of
such a diagram and then show that the totalization agrees with the homotopy
limit in the following two cases: (1) the complexes of sheaves of $\mathcal
O$-modules on the \v{C}ech nerve of an open cover of a ringed space $(X,
\mathcal O)$; (2) the complexes of sheaves on the simplicial nerve of a
discrete group $G$ acting on a space. The explicit models we obtain in this way
are twisted complexes as well as their $D$-module and $G$-equivariant versions.
As an application we show that there is a stack of twisted perfect complexes.
