In the mid 90's Eliashberg and Thurston proved that (all but one, sufficiently
smooth) foliations on 3-manifolds could be C^0- approximated by contact
structures. This result has been very useful in understanding contact
structures and in applications to low dimensional topology, but their ground
breaking work left many tantalizing unanswered questions. For example, do all
contact structures come as deformation of foliations? Can the C^0 in Eliashberg
and Thurston's result be improved? Can approximation be replaced by deformation?
How does the geometry of the foliation affect the geometry of the approximating
contact structure? The focus of this talk with be to place these questions in
the context of the 'space of plane fields' on a 3-manifold and then give, at
least partial, answers to these and other questions.