Based on the consideration to the layer structures of fuzzy lattices and the level topologies of L-fuzzy topological spaces, a connectedness is dened for an arbitrary L-fuzzy set in this paper. This denition reects the degree of connectivity of an L-fuzzy set. It is shown that the connectedness of L-fuzzy topological spaces is an L-good extension, multiplicative and preserved under continuous L-valued Zadeh functions, and that the inverse limit of continuums is a continuum. General L-fuzzy intervals and H-intervals are dened and connectedness of them are proved. It is also shown that the connectedness of an L-fuzzy topological space is equivalent to the connectedness of its induced I (L)-fuzzy topological space.