  We show some generic (robust) properties of smooth surfaces immersed in the
real 3-space (Euclidean, affine or projective), in the neighbourhood of a {\em
godron} (term due to R.Thom): an isolated parabolic point at which the (unique)
asymptotic direction is tangent to the parabolic curve. With the help of these
properties and a projective invariant that we associate to each godron we
classify the godrons and present all possible local configurations of the
flecnodal curve at a generic swallowtail in $R^3$. We present some global
results, for instance: {\em A closed parabolic curve bounding a hyperbolic disc
has a positive even number of godrons, and the flecnodal curve lying in that
disc has an odd number of transverse self-intersections.}.
