What are the pods with the sharp trailing edges underneath the wings of large airliners, as shown in the image below?
My best guess would be fuel tanks
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They are Anti-shock bodies.
In the transsonic speed range (above about Mach 0.7), aircraft drag is governed by Whitcomb area rule, which basically says that to minimize drag, the aircraft cross-sectional area must change as smoothly as possible, independent of its actual shape. It is somewhat counter-intuitive, but well established.
Compare this Junkers patent drawing (via Wikipedia):
which shows various combinations of positioning engines to adopt the area rule.
For the usual aircraft design the cross-section increases over the engines that hang ahead of the wing leading edge and then the thick part of the wing and wing box (the thicker part of fuselage where wings connect), but the wing tips are too thin and the cross-sectional area would decrease too quickly, so something needs to be attached to the trailing edge to make the reduction of cross-sectional area smoother. The anti-shock bodies are usually conveniently combined with flap actuator and track fairings.
There are however some aircraft that have anti-shock bodies combined with other functions, e.g. Tu-134 (and many other designs by Tupolev Design Bureau) retracts landing gear in its anti-shock bodies (which are just one large on each side):
The flap actuator and track fairings exist on all aircraft that have flaps that extend aft as well as down (Fowler flaps and their double-slotted variants), but on slower planes they are much thinner. Compare e.g. ATR-72, which does not need to be area-ruled with its maximum Mach 0.55.
They are not fuel tanks, the fuel tanks are inside the wings.
Those are covers for the flaps actuators: since the actuators and their rails protrude beneath the wing, without a cover they would greatly increase drag. The covers are more aerodynamic and thus produce overall less drag (at the expense of a slight increase in overall weight)
They are not anti-shock bodies nor fuel tanks.
The A320's flight manual (see slide 24) calls them "flap track fairing" since they are... well, fairings for the flap tracks and those fairings are just as big as needed to smoothly cover the flap tracks, nothing more nothing less.
The following picture (my work) shows one of these tracks of an A300 with fairing removed:
The straight yellow part (which I underlined in blue) is basically a rail fixed to the wing on which the two flanges (circled in green) slide toward the left bringing along the flaps; an endless screw (in red) actuates the sliding. The shape of the fairing can be understood from the background of the assembly: as visible, it is just as big as needed to enclose the structures, nothing more, nothing less.
So they don't carry out the duty of anti-shock bodies (or any other aerodynamic function) if not as a byproduct. Anti-shock bodies used to look like a bulge behind the trailing edge and aligned with the chord and protruding both over and under the wing, like in the Convair 990:
In modern jetliners that function is taken over by the quite big fairing located where the wing meets with the fuselage, in yellow on this A380 picture from Wikipedia:
Luckily enough these two pictures were taken from more or less the same angle and this makes the comparison easier.
There is a lot of misinformation here, and the top answer tries to combine concepts from two different applications. These 'pods' are flap track fairings, sometimes called trailing edge fairings, and their purpose is to cover and reduce drag over the heavy duty mechanisms that extend the flaps of an aircraft wing during high-lift operating regimes. See https://en.m.wikipedia.org/wiki/Aircraft_fairing
These pods are not anti-shock bodies, despite nearly all answers trying to use it as a buzzword. Anti-shock bodies are attached to the upper surface of the trailing edge (unlike flap track fairings, which are attached to the underside). Anti-shock bodies were used to reduce the shock that occurrs over the wing while traveling at transonic speeds. They were made obsolete with the advent of supercritical airfoils, and no longer found on modern aircrafts.