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Where are an airplane's engines located?
Short answer
On most airliners, the engines hang under the wings in nacelles attached to pylons; on regional and business jets they sit on either side of the tail, and on fighter jets they're embedded in the fuselage. Light propeller planes carry the engine in the nose, with the position chosen for weight, maintenance and safety reasons.

Under the wings, the dominant solution
Since the Boeing 707, and especially since high-bypass turbofans grew larger in diameter, most airliners carry their engines hung under the wings in nacelles attached to a pylon. The main reason is structural: the weight of the engine, located ahead of and below the wing, counteracts part of the upward bending caused by lift and dampens flutter, allowing for lighter wings. The engine also sits at a convenient working height for mechanics, with a low platform, far from the passenger cabin, and gets clean air ahead of it, undisturbed by the fuselage. If a turbine disintegrates, the fragments fly out in a plane that certification rules require engineers to study, but the rest of the vital structure remains relatively protected.
At the tail: quiet and a clean wing
Jets with engines mounted at the rear of the fuselage (Caravelle, DC-9, MD-80, CRJ and almost all business jets) pursue a different goal: a wing free of nacelles, with full-span flaps and greater aerodynamic efficiency, plus a quieter cabin, since the noise is left behind most of the seats. In exchange, the rear fuselage must be reinforced, the center of gravity shifts backward, and the horizontal stabilizer usually moves up into a T-tail configuration to escape the jet blast. Classic-era trijets (727, DC-10, L-1011) added a third engine at the base of the fin or inside it, with an S-duct in the case of the 727 and the L-1011.
Inside the fuselage: fighters and propeller planes
In fighter jets, the engine is embedded in the fuselage, with air intakes on the sides or under the nose. What is gained is a minimal frontal area, lower drag at high speed, a more contained radar signature, and the ability to place the weight near the center of gravity for maneuverability. Light propeller aircraft carry the engine in the nose, right in front of the firewall: it's the simplest solution, balancing the weight of the cabin and tail. Twin-engine propeller planes and regional turboprops mount them in nacelles on the leading edge, where the propeller sweeps the wing and improves lift at low speed.
Why there is no single answer
The final position results from a trade-off between structural weight, aerodynamic drag, cabin noise, risk of runway debris ingestion, ease of maintenance, and space for propellers or nozzles. Over the last twenty years the balance has tilted toward low wings with large-diameter turbofans, to the point that the Boeing 737 MAX and the A320neo had to redesign the landing gear and pylon to accommodate engines much wider than those of their predecessors.
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