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This comment to a question about underwing vs rear-mounted engines notes the existence of a rarely used third option: overwing engines.
There seems to be very few planes using those. What are the disadvantages that makes it so unpopular, and what are the advantages that made it worth building those few planes?
The biggest reason is the negative (or spoiling) effect of having a body "in the way" and disrupting the flow field above the wing. A wing makes lift by redirecting a very large package of air downward (actually, up a bit, then down a lot) as the wing moves along, and most of the package of redirected air (which ends up as downwash) is above the wing. (Otherwise, the wing would just be an air deflector and a flat sheet of plywood would do the job.) Disruptions to this upper flow have to be minimized.
Look at any tactical fighter, like a Skyraider, festooned with rockets and bombs and tanks under the wing almost across the entire span. The disruptive effects of all those shapes on the underside have minimal effect other than parasitic drag. Put them on top and they act like spoilers and the airplane would never get off the ground.
On airplanes that do put the engines on top, either the engine is so far aft that it minimizes the effect and it might was well be mounted on tail pylons (like the Honda Jet), or the engines are mounted way forward to minimize the effect, plus gain from having the high velocity fan stream across the top, or it's propeller driven and the propeller flow field mitigates the effect, or, it was done anyway and the design was unsuccessful.
The main advantages are keeping the engines clear of ground debris to avoid FOD and providing additional lift for STOL operations, as can be seen in the An-72.
These requirements are rare for civilian airliners operating out of modern airports, and are more common for military transport types and regional jets operating in remote areas.
The greatest disadvantage is the complicated maintenance, which requires stairs to even begin to inspect the engine. Furthermore, with the proliferation of composite structures in aviation, it is desirable to avoid the risk of a mechanic dropping a wrench on such a surface, and having technicians regularly working over the wing is a surefire way to end up replacing panels due to tool damage.
Another reason overwing engines are not favored for passenger planes is that instead of shielding the passengers from engine noise by the wing (as in the case of underslung engines) you have the noise source(s) on the same side of the wing as the passengers, which makes for a noisier cabin.
In addition, suspending an engine on a pylon under the wing allows the engine to be shed off by gravity in case of an inextinguishable fire that destroys the engine. It's not clear you could exploit this effect with engines mounted above the upper surface of the wing.
The reason why engines are mounted under the wing is easy maintenance access and easy visual inspection. When they are low to the ground you can detach the engines with relative ease compared to lifting them up from above the wing. This makes maintenance inspections quicker and thus cheaper.
During the walk around pilots can easily look into the engine intake, even grab the fan blades and turn them. You can have visual gauges like an oil level indicator at eye level.
In case of a fire hot burning oil doesn't drop down onto the wing and in case of an uncontained engine failure (e.g. turbine disk rupture) you at least have the wing in the way of the debris before it hits the cabin.
