  We introduce moduli spaces of colored graphs, defined as spaces of
non-degenerate metrics on certain families of edge-colored graphs. Apart from
fixing the rank and number of legs these families are determined by various
conditions on the coloring of their graphs. The motivation for this is to study
Feynman integrals in quantum field theory using the combinatorial structure of
these moduli spaces. Here a family of graphs is specified by the allowed
Feynman diagrams in a particular quantum field theory such as (massive) scalar
fields or quantum electrodynamics. The resulting spaces are cell complexes with
a rich and interesting combinatorial structure. We treat some examples in
detail and discuss their topological properties, connectivity and homology
groups.
