Several variants of the edit distance problem with block deletions are considered. Polynomial time optimal algorithms are presented for the edit distance with block deletions allowing character insertions and character moves, but without block moves. We show that the edit distance with block moves and block deletions is NP-complete (Nondeterministic Polynomial time problems in which any given solution to such problem can be verified in polynomial time, and any NP problem can be converted into it in polynomial time), and that it can be reduced to the problem of non-recursive block moves and block deletions within a constant factor.