Hydraulic drive has been used in the past, mostly on larger
boats (see one link below). The main advantage is that the
engine can be mounted anywhere in the boat and the drive
unit can be turned in any direction. I suspect that
compared to an outboard or stern drive it's more costly and
less efficient.

http://www.peachment.co.uk/drive.asp

wrote:
OK, I admit I know nothing about outboards but I cannot help myself
when it comes to inventing weird stuff, So............
Why not replace that whole heavy outboard on the transom with an engine
on the boat that turns a hydraulic pump that turns a hydraulic motor
hung from the transom. I know nothing about losses in the gear train
of an outboard vs those in such a hydraulic system but mounting would
sure be simpler as would be tilt and trim. The engine could be a
conventional lightweight 4 cycle without all the problems with making
it light enough for being transom hung. For that matter, maybe even a
small diesel could work instead of a gas engine to turn the hydraulic
pump thus making it all much more fuel efficient.
Of course, somebody has probably already thought of all of this.