  In this article we define intersection Floer homology for exact Lagrangian
cobordisms between Legendrian submanifolds in the contactisation of a Liouville
manifold, provided that the Chekanov-Eliashberg algebras of the negative ends
of the cobordisms admit augmentations. From this theory we derive several exact
sequences relating the Morse homology of an exact Lagrangian cobordism with the
bilinearised contact homologies of its ends. These are then used to investigate
the topological properties of exact Lagrangian cobordisms.
