  We classify all order one invariants of immersions of a closed orientable
surface F into R^3, with values in an arbitrary Abelian group G. We show that
for any F and G and any regular homotopy class A of immersions of F into R^3,
the group of all order one invariants on A is isomorphic to G^\aleph_0 \oplus B
\oplus B where G^\aleph_0 is the group of all functions from a set of
cardinality \aleph_0 into G and B={x\in G : 2x=0}. Our work includes
foundations for the study of finite order invariants of immersions of a closed
orientable surface into R^3, analogous to chord diagrams and the 1-term and
4-term relations of knot theory.
