TITLE: Isomorphic fundamental groups result in homeomorphism?
QUESTION [9 upvotes]: I know that the fundamental group of homeomorphic spaces are isomorphic. Is the converse true? I mean, can we say the two spaces with isomorphic fundamental groups are homeomorphic?

REPLY [8 votes]: This is an addendum to Jason DeVito's answer; it concerns the current status of Borel conjecture: 
If two closed aspherical (i.e., with contractible universal covers) manifolds $M, N$ have isomorphic fundamental groups then they are homeomorphic. 
The strongest to date result towards Borel conjecture in dimensions $\ge 5$ is in the paper 
"The Borel Conjecture for hyperbolic and CAT(0)-groups" by Bartels and Lueck which can be found here. 
Their theorem states that Borel conjecture (in dimensions $\ge 5$) holds for CAT(0) and hyperbolic groups; this result includes the earlier results by Farrel and Jones on Borel conjecture for nonpositively curved manifolds. 
Furthermore, Borel conjecture holds in dimension 3 thanks to Perelman's solution of Thurston's Geometrization Conjecture. People usually are aware of the fact that Perelman proved Poincare conjecture in dimension 3, but his theorem proves much more than this. Here is how Borel conjecture in 3d follows from Perelman's work:
Perelman's result immediately implies that every closed 3-manifold $M$ with contractible universal cover is one of the following:

Hyperbolic. 
Seifert. 
Haken.

Borel conjecture for hyperbolic manifolds follows from Mostow rigidity theorem.  For Haken manifolds Borel conjecture was proven by Waldhausen (see here). For Seifert manifolds with contractible universal cover Seifert invariants (which completely determine topology of the manifolds) can be read off the fundamental group (my guess is that this was known to Seifert). 
Borel conjecture in dimension 4 is wide-open as far as I know.