I make no claims about any of this being authoritative, but it's the best I've been able to glean from several sources.
In the 1860s, one Guido (good name that) Panerai set up a wordshop in Florence making precision mechanical instruments. The workshops carried on into the 20th century as "Officine Panerai".
By the 1930s the workshops were making a range of wrist-worn instruments for frogmen; compasses, depth gauges and the like.
In 1935 Officine Panerai were comissioned by the Italian Navy to construct watches for use underwater, specifically ones that could be read underwater. After some inital prototypes Officine Panerai delivered it's first product diving watches in 1938. These were fitted with Rolex movements (and Rolex crowns and casebacks).
Officine Panerai progressively improved its designs, with increased luminosity of numerals and hands, and the now famous crown-locking device. Later models were fitted with an Angelus 8 day (marked "8 giorno") movement . The radioactive radium which gave such legibility to the Radiomir models was replaced with the distinctly less toxic compound "Luminor".
Things get a bit hazy here, but as far as I can see the equipment manufacturing end of Officine Panerai drifted out of business after the end of the Second World War.
In 1993 Officine Panerai started to reproduce the Luminor, and the Mare Nostrum chronograph, basically targetted at the collector's market. These were produced only in very limited numbers.
In 1997 Officine Panerai were acquired by the Vendôme group. Under the new bosses basically the same models were produced, still in small numbers. However in a clever marketing move, a few "unique editions" were produced each year, with an intriguing movement or case design.
This basically brings us to the present day. There seems to be no great rush to produce Officine Panerai watches in huge numbers. Collectors are snapping up vintage models and "pre - Vendôme", and Panerai have just launched their most exciting range of production watches and unique editions yet. It will be intriguing to see what happens in the next few years, but with a steady hand on the helm Panerai might well become one of the most collectable brands around, whilst still not being seen on every street corner.