  We prove that for any closed surface of genus at least four, and any
punctured surface of genus at least two, the space of ending laminations is
connected. A theorem of E. Klarreich implies that this space is homeomorphic to
the Gromov boundary of the complex of curves. It follows that the boundary of
the complex of curves is connected in these cases, answering the conjecture of
P. Storm. Other applications include the rigidity of the complex of curves and
connectivity of spaces of degenerate Kleinian groups.
