  Lagrangian cobordisms are three-dimensional compact oriented cobordisms
between once-punctured surfaces, subject to some homological conditions. We
extend the Le-Murakami-Ohtsuki invariant of homology three-spheres to a functor
from the category of Lagrangian cobordisms to a certain category of Jacobi
diagrams. We prove some properties of this functorial LMO invariant, including
its universality among rational finite-type invariants of Lagrangian
cobordisms. Finally, we apply the LMO functor to the study of homology
cylinders from the point of view of their finite-type invariants.
