Hello, Which manifolds in dimension five admit contact structures? I am not too familiar with the contact realm so any references to look at would be much appreciated.
Let M^{5} be a closed and oriented. A contact form α gives a 4plane distribution with symplectic form dα, reducing the structure group of TM to U(2)×1; such a reduction is called an almost contact structure, and it exists iff the integral third StiefelWhitney class is zero (Gray, "Some global properties of contact structures", MR0112161). Equivalently, M is almost contact iff w_{2}(M) lifts to an integral cohomology class. The existence problem for actual contact structures is much harder, though. Contact structures are known to exist in at least the following cases:
I'm not an expert, but I don't think there are any almost contact 5manifolds which are known to not be contact. Update (7/25/15): The problem has been completely solved since my original answer in 2011. For 5manifolds, Casals, Pancholi, and Presas and Etnyre proved that every homotopy class of almost contact structure contains a contact structure. This was then generalized to all odd dimensions by Borman, Eliashberg, and Murphy, who gave a definition of overtwistedness in higher dimensions and showed that every homotopy class of almost complex structure contains a unique overtwisted contact structure up to isotopy. 

