Typical cranes mounted onto barges but also barges with cranes on top, as well as crane ships.
Crane ships typically are self-propelled and the cranes are mounted on the vessel.
The Heerema ‘SLEIPNIR’
Looking at one example of a huge, if not the largest crane ship, the Heerema “Sleipnir” with two cranes rotating on 30m diameter swing bearings. The ship has a self-balancing system and has 8 balancing reservoirs which each can take in 3,000 cubic meters each. The reservoirs can be flooded ‘naturally’, but can also be filled and/or emptied by huge pumps.
Furthermore, a gallery with various floating cranes. You click on the pictures to see the full-sized file