**Models for the moduli spaces of Riemann surfaces**

We consider Riemann surfaces *F* the boundary of which is partitioned
into *m* outgoing and *n* incoming boundary circles; we assume
that *m*, *n* > 1. For the moduli space
**M**_{g}(*m*, *n*) of all such *F*
of genus *g* ≥ 0 a configuration space model
**Rad**_{g}(*m*, *n*) is developed: it consists
of configurations of 2*g* + *m* pairs of radial slits
on *n* annuli satisfying a certain combinatorial condition to guarantee
exactly *m* outgoing circles. The space
**Rad**_{g}(*m*, *n*) is homotopy-equivalent to
**M**_{g}(*m*, *n*).

The basic tool is to describe the conformal structure of *F* by the
gradient flow of a uniquely determined harmonic function with constant
boundary values. The critical flow lines dissect *F* into annuli with
slits.

The space **Rad**_{g}(*m*, *n*) is a non-compact
manifold. Its closure provides a finite cell complex, which can be used
for homological calculations. Furthermore, the family of spaces
**Rad**_{g}(*m*, *n*) form an operad, and we
discuss various other spaces connected to this structure.

*Carl-Friedrich Bödigheimer*