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If the center of gravity of an airplane is too far aft, it will become more unstable

Iif the CG is too far forward, then drag will increase due to increased angle of attack.

But what are the benefits to operating with the CG near the aft, or forward, CG limits? In what situations would one choose to do so?

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Large aircraft typically have an optimum c.g. recommended. That c.g. is in terms of percent of mean aerodynamic chord, often expressed as %MAC. For the 747-100s and -200s, the figure is 26.6 %MAC for the zero fuel weight c.g. – Terry Jan 9 '14 at 0:29
up vote 11 down vote accepted

A CG near the aft reduces drag by reducing the necessary downforce that has to be supplied (usually) by the tailplane.
This is also a point for competition gliding (more often than not you have water ballast in the vertical tailplane to compensate for those love handles). Aerobatic gliders sometimes feature aft lead ballast to enable maneuvers like tailspins which are not possible with a forward CG.

Aft CGs come with reduced stability, which can be a severe flight-safety concern (e.g. in case of unrecoverable spins).
Some fighter aircraft feature inherently unstable aerodynamics (see super-maneuverability) which offer great agility (e.g. turning rates), but need constant input by a flight constrol system (and woe if that fails, e.g. via Pilot Induced Oscillation).

A forward CG results in greater flight stability and reduced danger of sudden stalling, but also increased drag (because of the increased downforce required from the tailplane). In extreme cases this can limit the maneuverability as well and cause a pancake landing (the tailplane can only produce so much lift/downforce before itself stalls, at which point it is no longer holding your nose up).

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A tailplane doesn't really produce lift because it's more like an inverted wing (unless negative lift is considered lift). The further aft the CG, the less the tailplane has to push the tail down (causing drag). That's why a canard design is so much more efficient because all wings produce "positive" lift. – Philippe Leybaert Jan 7 '14 at 22:37
@PhilippeLeybaert I tweaked the answer a bit to use the term "downforce" which is what most of us in the US are used to seeing, but technically (aerodynamically) most if not all tailplanes are producing "lift" to control pitch - it's just directed downward because the airfoil is "upside-down" compared to what we're used to in a wing :-) More downforce means more (negative) lift, and thus more drag... – voretaq7 Jan 7 '14 at 22:43
I stand corrected dor missing out on the negative sign, but fortunately the drag part is correct :D – yankeekilo Jan 7 '14 at 22:44
@PhilippeLeybaert: Some horizontal stabilizers are designed to generate lift or tail down force, depending on the regime of flight and the CG.... It isn't always tail down. – Lnafziger Jan 7 '14 at 23:00
@LNafziger Yes, you're correct but it's almost always tail down :-) But I felt it was important to point out that the tailplane normally doesn't contribute to the total lift as it cancels out some of the lift of the main wings (reducing total lift). – Philippe Leybaert Jan 7 '14 at 23:10

The main benefit of operating with an aft CG, as mentioned, is that it can measurably reduce drag. This happens because less tail down force is required to maintain level flight in a given configuration. As a result, you can burn less fuel to maintain the same speed (or increase your speed for no additional fuel burn) compared to a forward CG.

As long as you ensure you remain within your aircraft's CG envelope, you can test this out by having a passenger shift their seat aft while in cruise.

There are, of course, some safety concerns. @yankeekilo's answer covers this thoroughly, but if nothing else it should be reiterated that an aft cg is inherently less stable.

A forward CG makes recovery from stalls much easier, and in some cases can make stalls much harder to achieve because the moment-arm of your aircraft's CG changes. This change can leave the tailplane with insufficient input during slow flight, which is why you may never reach stall in the first place; it also makes landings dangerous as there may not be enough control range to flare. If this happens, you'll wheelbarrow or porpoise the aircraft.

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We call that a monday-(week day of your choice) landing in German, depending on how often you hear the wheels squeak :D – yankeekilo Jan 7 '14 at 20:06
@yankeekilo Hah, that's awesome. – egid Jan 7 '14 at 21:11

Advantages of forward CG position:

  • Higher stability, so less attention is needed to keep the trimmed aircraft straight and level.
  • If the aircraft can be made to spin, the spin is easier to end.
  • For taildraggers: The tail can be lifted up earlier during the take-off run.
  • For tricycle landing gear aircraft: Better stability on the ground.

Well, that's about it!

Advantages of rear CG position:

  • Less trim drag, because less downforce needs to be created at the tail and compensated by the wing.
  • Better maneuverability and lower control forces, because less control surface deflection is needed for the same excursion from the trimmed state.
  • Lower minimum speed, which translates into lower take-off and landing distances.
  • For taildraggers: More stability on the ground against tipping over.
  • For tricycle landing gear aircraft: Lower rotation speed possible at takeoff.

Note that shifting the CG more aft, behind the neutral position, will make the aircraft statically unstable and will increase trim drag again, because now the tail will operate at abnormally high lift coefficients which come with high induced drag.

The typical application for forward CG positions is a manually controlled long range aircraft which will fly in gusty weather. Then it will be beneficial to have high stability.

The typical applications for rear CG positions are performance- or maneuverability-optimized aircraft like gliders or aerobatic airplanes.

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