june 1 if everything works
the earlier the better. i turned mine in july 1 (last year we couldn't submist until about june 18 due to problems with web servers).
i got early interviews at some schools. i even interviewed with early decision applicants, which suprised the interviewer (bonus for me!). i got in about 3 weeks after they found out their decisions.
it can only help you to get in early.
also the secondaries are the true pains in the buttocks, you have to do one for each school. sometimes you can recycle essays. sometimes you cannot.