  Hamiltonian flows on compact surfaces are characterized, and the topological
invariants of such flows with finitely many singular points are constructed
from the viewpoints of integrable systems, fluid mechanics, and dynamical
systems. Though various fluid phenomena are modeled as flows on the plane, it
is not obvious to determine if the flows are Hamiltonian, even the singular
point set is totally disconnected and every orbit is contained in a straight
line parallel to the $x$-axis. In fact, there are such non-Hamiltonian flows on
the plane. On the other hand, this paper topologically characterizes
Hamiltonian flows on unbounded surfaces and constructs their complete invariant
under a regularity condition for singular points. In addition, under finite
volume assumption, Hamiltonian flows on unbounded surfaces can be embedded into
those on compact surfaces.
