(1) as derivations over an algebra of smooth functions,
(2) as ``vectors attached to points'', i.e. directions of curves.

Both notions are in fact equivalent, and while (1) is more abstract, it is easily extended to arbitrary metric spaces (with smooth functions substituted by Lipschitz ones), thus leading to the identification ``vector fields = one-dimensional currents''. It will be shown that even in such a generality vector fields corresponding to normal currents have an underlying structure, provided by rectifiable curves (i.e. they can be viewed, in a sense, like (2)). Several consequences of this fact will be discussed.
http://cvgmt.sns.it/seminar/305/