I will discuss joint work with Olga Plamenevskaya studying symplectic fillings of links of certain complex surface singularities, and comparing symplectic fillings with complex smoothings. We develop characterizations of the symplectic fillings using planar Lefschetz fibrations and singular braided surfaces. This provides an analogue of de Jong and van Straten's work which characterizes the complex smoothings in terms of decorated complex plane curves. We find differences between symplectic fillings and complex smoothings that had not previously been found in rational complex surface singularities.