Question: I have the following question:

For a given two-dimensional Riemann surface $C$,

1. Is there a way to classify all topologically distinct
three-dimensional compact manifolds $M$ whose boundary is $C$,
i.e., $\partial M =C$?
2. Is there always a three-dimensional compact manifold
$M$ such that $\partial M =C$ and is contractible?
3. If there exists $M$ that satisfies condition 2,
is it unique? If not unique, can the set of such manifolds
be classified in a nice way?

Thank you in advance!

Answer: 
1. When you say "Riemann surface", do you mean "topological surface"? Does the Riemann surface structure have any significance?

I assume below that you mean "two-manifold"

1. Well, any three manifold contains a handlebody of your favorite genus, so this question is at least as hard as classifying three-manifolds (which is possible, by Thurston-Perelman).
2. No, by the half-lives half-dies theorem.

3.Vacuous, because of 2.
For (3), here is a nice counterexample if we ignore contractibility (taken from an article from *The American Mathematical Monthly*):

<http://www.jstor.org/stable/pdfplus/2695643.pdf>

Hopefully someone can embed the image (Figure 13). It is two topologically distinct manifolds with the torus as a boundary... one being a solid [knot] torus, and the other being a 'cylinder' with a knotted inner-hole.