Are there advantages when operating with CG near forward or aft limits?
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).