  Let C be the contact structure naturally induced on the lens space L(p,q) by
the standard contact structure on the three--sphere. We obtain a complete
classification of the symplectic fillings of (L(p,q),C) up to
orientation-preserving diffeomorphisms. In view of our results, we formulate a
conjecture on the diffeomorphism types of the smoothings of complex
two-dimensional cyclic quotient singularities.
