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Symplectic fillings of standard tight contact structures on lens spaces areunderstood and classified. The situation is different if one considersnon-standard tight structures (i.e. those that are virtually overtwisted), forwhich a classification scheme is still missing. In this work we use differentapproaches and employ various techniques to improve our knowledge of symplecticfillings of virtually overtwisted contact structures. We study curvesconfigurations on surfaces to solve the problem in the case of a specificfamily of lens spaces. Then we give general constraints on the topology ofStein fillings of any lens space by looking at algebraic properties of integerlattices and at geometric slicing of solid tori. Furthermore, we try to placethese manifolds in the context of algebraic geometry, in order to determinewhether Stein fillings can be realized as Milnor fibers of hypersurfcesingularities, finding a series of necessary conditions for this to happen. Inthe concluding part of the thesis, we focus on the connections between planarcontact 3-manifolds and the theory of Artin presentations.

Symplectic fillings of virtually overtwisted contact structures on lens spaces