Where were the knights hospitaller based?

They were based in Jerusalem until Jerusalem fell in 1187, Then Acre until it too fell in 1291. From Acre they settled for a time in Cyprus, in 1309 they captured Rhodes and settled there until they were defeated in 1522 by Suleiman. In 1530 the order was granted the island of Malta where they resided until it was captured by Napolean in 1798 after which in 1834 they settled in rome where they still reside today