  This note is but a research announcement, summarizing and explaining results
proven and detailed in forthcoming papers. When one studies families of objects
over curves, and the objects are parametrized by a Deligne-Mumford stack M,
then the families are equivalent to morphisms of curves into M. In order to
have complete moduli for such families, one needs to compactify the stack of
stable maps into M. It turns out that in the boundary, the curve must acquire
extra structure and you'd better read the paper to see what that structure is.
  Applications to fibered surfaces (see math.AG/9804097), admissible covers,
and level structures are discussed.
