There is something called an aspherical manifold. This is a closed manifold whose universal cover is contractible. In particular any aspherical manifold $M$ is a model for $B\pi$ where $\pi=\pi_1(M)$. There are many examples of aspherical manifold, for example any closed manifold of negative sectional curvature (e.g. hyperbolic manifolds) are aspherical by the Theorem of Cartan-Hadamard, that the exponential map is a then covering map.
Now there is a beautiful conjecture due to Borel (the Borel Conjecture) which states that any two aspherical manifolds $M$ and $N$ with isomorphic fundamental group $\pi$ are homeomorphic. Even more the conjecture predicts that any homotopy equivalence $f: M \to N$ is homotopic to a homeomorphism.
Recently there has been a lot of work concerning this conjecture due to a stronger conjecture, the Farrell-Jones Conjecture. This is a conjecture about algebraic $K$ and $L$ theory of group rings.
The Farrell-Jones Conjecture for $K$ and $L$ theory together imply the Borel Conjecture.
Moreover the Farrell-Jones Conjecture has been proven for a quite large class of groups including hyperbolic groups, $CAT(0)$-groups and many more.
A lot of work on the Farrell-Jones Conjecture is due to Wolfgang Lück ( professor at Bonn university ) and you might want to look at some of his survey articles concerning these kinds of questions. You can find them on his homepage, http://www.math.uni-bonn.de/ag/topo/members for a link to that.
Moreover I should mention that there is a theorem called "Mostow rigidity" which proves the Borel conjecture for hyperbolic manifolds, and this is much older than the work on Farrell-Jones Conjecture.