  Symplectic fillings of standard tight contact structures on lens spaces are
understood and classified. The situation is different if one considers
non-standard tight structures (i.e. those that are virtually overtwisted), for
which a classification scheme is still missing. In this work we use different
approaches and employ various techniques to improve our knowledge of symplectic
fillings of virtually overtwisted contact structures. We study curves
configurations on surfaces to solve the problem in the case of a specific
family of lens spaces. Then we give general constraints on the topology of
Stein fillings of any lens space by looking at algebraic properties of integer
lattices and at geometric slicing of solid tori. Furthermore, we try to place
these manifolds in the context of algebraic geometry, in order to determine
whether Stein fillings can be realized as Milnor fibers of hypersurfce
singularities, finding a series of necessary conditions for this to happen. In
the concluding part of the thesis, we focus on the connections between planar
contact 3-manifolds and the theory of Artin presentations.
