So I was reading N.G.L. Hammond's 'A History of Greece to 322 B.C.' and it has gotten me very interested in various Bronze Age civilisations. Especially since Hammond's book was written in the 1950s so I am guessing that the archaeological and historical knowledge of the period has moved a long way since he was writing. So I was wondering if there are any good books covering modern scholarship on the subject.
The AskHistorians reading list has most of the best introductory works on the Babylonians, Assyrians, Hittites, Elamites, Egyptians, etc.
I haven't gotten around to updating the Aegean reading list yet, but see my responses here for Mycenaean reading suggestions. Minoans by J. Lesley Fitton is a very good introduction to the Minoans.
For the Levant, see my response to Help/suggestions on what books to read pertaining to ancient Syrian history. To that, one can add Jonathan Tubb's Canaanites and Karageorghis' Early Cyprus: Crossroads of the Mediterranean.