We consider the problem of describing Stein fillings of
contact 3-manifolds. We prove topological constraints such 4-manifolds
have to satisfy. Examples of contact 3-manifolds with infinitely many
fillings will be given. Finally we show infinitely many examples of
contact 3-manifolds with no fillings, even in a weaker sense.
In the course of the proofs we use various aspects of Seiberg-Witten
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