Is it possible to cover 3-dimensional space by a collection of lines, such that no two lines intersect and no two lines are parallel?  More precisely, does there exist a fibration of R^3 by pairwise skew lines?  We give some examples and provide a topological classification of these skew fibrations.  We continue with some recent results regarding contact structures on R^3 which are naturally induced by skew fibrations.  Finally, we discuss fibrations of R^3 which may contain parallel fibers, and discuss some structural results for such fibrations, as well as their relationship with contact structures.