  We prove that the mapping stack Map(Y,X) of topological stacks X and Y is
again a topological stack if Y admits a compact groupoid presentation. If Y
admits a locally compact groupoid presentation, we show that Map(Y,X) is a
paratopological stack. In particular, it has a classifying space (hence, a
natural weak homotopy type). We prove an invariance theorem which shows that
the weak homotopy type of the mapping stack Map(Y,X) does not change if we
replace X by its classifying space, provided that Y is paracompact topological
space. As an example, we describe the loop stack of the classifying stack BG of
a topological group G in terms of twisted loop groups of G.
