  In this paper we use Floer theory to study topological restrictions on
Lagrangian embeddings in closed symplectic manifolds. One of the phenomena
arising from our results is ``homological rigidity'' of Lagrangian
submanifolds. Namely, in certain symplectic manifolds, conditions on low
dimensional topological invariants of a Lagrangian (such as its first homology)
completely determine its entire homology.
  We also develop methods for studying Hamiltonian displacement of Lagrangian
submanifolds and its relations to the topology of Lagrangians. Finally we
present some new results on Lagrangian intersections.
