Gauge theoretical methods in the classification of non-Kaehlerian surfaces

The classification of class VII surfaces is a very difficult classical problem in complex geometry. It is considered by experts to be the most important gap in the Enriques-Kodaira classification table for complex surfaces. The standard conjecture concerning this problem states that any minimal class VII surface with $b_2>0$ has $b_2$ curves. By the results of Kato, Nakamura and Dloussky/Oeljeklaus/Toma, this conjecture (if true) would solve this classification problem completely. We explain a new approach (based on techniques from Donaldson theory) to prove existence of curves on class VII surfaces, and we present recent results obtained using this approach.