We study in this article topological structure of
divergence-free vector fields on general two-dimensional manifolds.
We introduce a new concept called block structural stability
(or block stability for simplicity) and
prove that the block stable divergence-free vector fields form
a dense and open set. Furthermore, we show that a block stable
divergence-free vector field, which we call a basic vector field,
is fully characterized by a nice and simple structure,
which we call block structure. The results and ideas
presented in this article have been applied to studies on structure and its
evolutions of the solutions of
the Navier-Stokes equations; see [4, 9, 10].