A geodesic current is a measure on the space of unoriented geodesics of the universal cover of a surface S (genus > 1) that is invariant under the action of the deck transformations. Geodesic currents were invented to generalize Thurston's measured laminations, and provide a natural completion of the Teichmuller space of S, T(S). Using geodesic currents, we can view T(S) as a "submanifold of a certain infinite dimensional analog of the hyperbolic space." Our main tool in our investigations will be a generalization of the geometric intersection number of curves on S.