  We define the notion of {\em classifying space} of a topological stack and
show that every topological stack \X has a classifying space X which is a
topological space well-defined up to weak homotopy equivalence. Under a certain
paracompactness condition on \X, we show that X is actually well-defined up to
homotopy equivalence. These results are formulated in terms of functors from
the category of topological stacks to the (weak) homotopy category of
topological spaces. We prove similar results for (small) diagrams of
topological stacks.
