  Let $\ccM_{g,[n]}$, for $2g-2+n>0$, be the stack of genus $g$, stable
algebraic curves, endowed with $n$ unordered marked points.
  Looijenga introduced the notion of Prym level structures in order to
construct smooth projective Galois coverings of the stack $\ccM_{g}$.
  In \S 2 of this paper, we introduce the notion of Looijenga level structure
which generalizes Looijenga construction and provides a tower of Galois
coverings of $\ccM_{g,[n]}$ equivalent to the tower of all geometric level
structures over $\ccM_{g,[n]}$.
  In \S 3, Looijenga level structures are interpreted geometrically in terms of
moduli of curves with symmetry. A byproduct of this characterization is a
simple criterion for their smoothness. As a consequence of this criterion, it
is shown that Looijenga level structures are smooth under mild hypotheses.
  The second part of the paper, from \S 4, deals with the problem of describing
the D-M boundary of level structures. In \S 6, a description is given of the
nerve of the D-M boundary of abelian level structures. In \S 7, it is shown how
this construction can be used to "approximate" the nerve of Looijenga level
structures. These results are then applied to elaborate a new approach to the
congruence subgroup problem for the Teichm\"uller modular group.
