Early attempts
There are ancient carved images of animal skins filled with air being used as one-man floats to cross rivers. They were inflated by mouth. Sometimes these images have been wrongly described as ancient scuba.
In 1839 the Duke of Wellington tested the first inflatable pontoons.

Rubber arrives
In 1900 to 1910 the development of rubber manufacturing enabled attempts at producing circular rubber inflatable boats: similar to modern coracles. These were only usable as rafts and could be propelled only by paddling, and they tended to crack at seams and folds due to imperfect manufacture of the rubber.

The Titanic and WWI
With the loss of the RMS Titanic in 1912, and World War I losses of ships to submarine-launched torpedoes, the need for inflatable boats was plain.
One cause of the loss of life on the Titanic was the lack of lifeboats. Even if every lifeboat had been completely filled with passengers and crew, there would have been no way to rescue more than half of all the people on board. The first SOLAS treaty was designed to avoid such a disaster happening again. One of its provisions was to ensure that vessels had enough lifeboats to provide every person aboard the vessel with a place. Putting this rule into effect was not difficult with cargo ships: they had small crews and plenty of deck space. Passenger ships had to stack lifeboats on top of each other to able to carry enough to accommodate the large number of passengers and crew. Warships also had large crews and little deck space.
Between the two World Wars, Goodyear found a way to join rubber to other materials. They made life rafts of square-shaped inflated rubber tubes with a rigid floor. Such rafts were to be stacked vertically aboard warships, usually standing on deck and leaning against deck-houses. But conservative thinking from navies held back this new idea.
Pierre Debroutelle's 1937 design was the first known to have its inflatable tube in a U-shape. It was the first boat of its kind to be certified by the French Navy. Its added wooden transom was patented on 10 August 1943. This version was the predecessor of today's inflatable sports and pleasure boats.

World War II
World War II changed everything. Submarine warfare in the Battle of the Atlantic led to casualties among warships and merchant ships.
US warships began using rubber life rafts. Since the rubber was much higher quality than 35 years before, the inflatable returned, but this time it was boat-shaped.
In military use inflatable boats were used to transport torpedoes and other cargo. They also allowed troops to make landings in shallow water, and their compact size and storability made overland transport possible.
One of the models, the Zodiac brand inflatable boat, grew to be popular with the military and contributed significantly to the rise of the civilian inflatable boat industry, both in Europe and in the United States. After World War II, surplus inflatable boats were sold to the public. A version of this boat has been adapted by the Marine Mammal Center for use in rescuing injured marine mammals at sea.

See also: LCRS
Modern inflatables
A modern Hypalon inflatable boat with rigid wooden floorboards and transom and an inflatable keel. Powered by a 12 Volt electric trolling motor
Inflatable life rafts were also used successfully to save crews of aircraft that ditched in the sea; bombing, naval and anti-submarine aircraft flying long distances over water being much more common from the start of WWII. The PBY Catalina made by Consolidated Aircraft and Canadian seems to have been the first aero plane to have had an inflatable life boat aboard, first as optional, later as standard equipment. A later version of that inflatable was pressurized by a gas cylinder rather than by mouth. A wire connected to the plane opened the cylinder valve in the inflatable after the life raft was thrown into the water.
Until the middle 1950s inflatables were still rafts in civilian use, hand paddled but the outboard motor came into use in the early 1950s. (The outboard motor was invented in 1909 by Ole Evinrude.)
Also in the 1950s, the French Navy officer and biologist Alain Bombard was the first to combine the outboard engine, a rigid floor and a boat shaped inflatable. The former airplane-manufacturer Zodiac built that boat and a friend of Bombard, the diver Jacques-Yves Cousteau began to use it, after Bombard sailed across the Atlantic Ocean with his inflatable in 1952. Cousteau was convinced by the shallow draught and good performance of this type of boat and used it as tenders on his expeditions.

The inflatable boat was so successful that Zodiac lacked the manufacturing capacity to satisfy demand. In the early 1960s, Zodiac licensed production to a dozen companies in other countries. In the 1960s, the British company Humber was the first to built Zodiac brand inflatable boats in the UK.
Some inflatables have inflated keels whose V-shape help the hull move through waves reducing the slamming effect caused by the flat hull landing back on the surface the water after passing over the top of a wave at speed.