I will construct infinite families of Rohlin invariant one Brieskorn homology spheres bounding rational balls (a joint work with Kyle Larson). I will then switch over discussing homology spheres bounding contractible manifolds. A contractible manifold W can fail to be a cork, either if there is no exotic involution on its boundary, or W fails to be Stein. We will give examples of Stein non-corks, and non-Stein non-loose corks. It is still an open question whether there are loose-corks that can not be corks (this part is a joint work with Danny Ruberman).