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This question has a variety of correct answers depending on various factors.
Many new vehicles shut off the fuel when the engines is being turned at an RPM higher than idle with your foot off the accelerator pedal. I've seen systems that do this since the 1970's! So on long down hill slopes the only difference between shutting off the engine and coasting in neutral and coasting in gear is the former causes more brake wear.
On older cars, especially carbureted ones, I've seen guys rig up solenoid valves and such that shut off fuel by means of a switch. Of course the hard part with a carburetor is draining the bowl. If you're going over a mountain pass and you shut it off too soon you can burn too lean on the incline and blow a piston! But if you do it too late coasting uses such little fuel that your engine will just be finishing up the bowl of fuel by the time you reach the bottom of the pass.
On small slopes it's most efficient to put it in neutral and turn off the engine a ways before the downhill slope. That way your car is going slow enough you won't need any braking by the time you've hit the bottom of the hill. Then you just keep coasting until you slow back down to your normal speed.
As far as automatic transmissions are concerned I knew a guy who ruined a few transmissions coasting them in neutral with the engine off.
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