TITLE: Examples of odd-dimensional manifolds that do not admit contact structure
QUESTION [15 upvotes]: I'm having an hard time trying to figuring out a concrete example of an odd-dimensional closed manifold that do not admit any contact structure. 
Can someone provide me with some examples?

REPLY [6 votes]: Any closed contact manifold is the boundary of an almost complex manifold. (This is written down in detail here.)
In particular, any manifold which is not null-cobordant, cannot carry a contact structure. (The Wu manifold $SU(3)/SO(3)$ is the simplest such manifold.)