  It is well known that not every combinatorial configuration admits a
geometric realization with points and lines. Moreover, some of them do not even
admit realizations with pseudoline arrangements, i.e., they are not
topological. In this paper we provide a new topological representation by using
and essentially generalizing the topological representation of oriented
matroids in rank 3. These representations can also be interpreted as curve
arrangements on surfaces.
  In particular, we generalize the notion of a pseudoline arrangement to the
notion of a quasiline arrangement by relaxing the condition that two
pseudolines meet exactly once and show that every combinatorial configuration
can be realized as a quasiline arrangement in the real projective plane. We
also generalize well-known tools from pseudoline arrangements such as sweeps or
wiring diagrams. A quasiline arrangement with selected vertices belonging to
the configuration can be viewed as a map on a closed surface. Such a map can be
used to distinguish between two "distinct" realizations of a combinatorial
configuration as a quasiline arrangement.
