There have only been a few aircraft built with forward-swept wings.
Most notably was the X-29, which you have pictured above.
Russia/USSR built one more recently in 1997, the Sukhoi Su-47.
One of the biggest claims as to the superiority or an advantage of a forward-swept wing was increased maneuverability.
However, in the case of the X-29, as noted in the Flying Qualities Evaluation of the X-29A Research Aircraft , The high pitch instability of the airframe led to wide predictions of extreme maneuverability.
This perception has held up in the years following the end of flight tests.
Air Force tests did not support this expectation.
For the flight control system to keep the whole system stable, the ability to initiate a maneuver easily needed to be moderated.
This was programmed into the flight control system to preserve the ability to stop the pitching rotation and keep the aircraft from departing out of control.
As a result, the whole system as flown (with the flight control system in the loop as well) could not be characterized as having any special increased agility.
It was concluded that the X-29 could have had increased agility if it had faster control surface actuators and/or larger control surfaces.
The Su-47 was indeed a highly maneuverable aircraft, capable of pulling 9 Gs, however, its immediate predecessor, the Sukhoi Su-37, was capable of pulling 10 Gs .
So it's not clear whether a forward swept wing has any real positive impact on maneuverability.
However, stall characteristics are very different.
Air tends to travel towards the rearmost end of the wing.
On a standard configuration (Rear-swept wing), this of course moves from the wing root to the wingtip.
On a forward-swept wing, however, this moves from the wingtip to the wing root.
As a result, the dangerous tip stall condition of a backwards-swept design becomes a safer and more controllable root stall on a forward swept design.
This allows full aileron control despite loss of lift, and also means that drag-inducing leading edge slots or other devices are not required.
With the air flowing inwards, wingtip vortices and the accompanying drag are reduced, instead the fuselage acts as a very large wing fence and, since wings are generally larger at the root, this improves lift allowing a smaller wing.
As a result maneuverability is improved, especially at high angles of attack.
At transonic speeds, shockwaves build up first at the root rather than the tip, again helping to ensure effective aileron control.
Because of the capability of the X-29 to have the tips of its wings bend down when at high angles of attack, the X-29 remained controllable at an angle of attack of 67 degrees.
However, thanks to modern advances in thrust vectoring, the F-22 Raptor is capable of sustaining an angle of attack over 60 degrees.
With inherent instability and problems with extra stresses on the airplane for only moderate or supposed but not observed gains in instability, it is unlikely we will see combat aircraft with forward swept wings any time in the near future.
Other technologies have enabled similar or enhanced performance without the need for novel concepts.
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Advantages: The wing spar carry-through can be placed aft of the cabin, so cabin height can be increased.
This is important for business-jet sized aircraft.
The boundary layer at the tips is not affected by the inner wing.
Controllability can be maintained up to stall.
Aeroelastic effects will increase control commands.
This makes for a very responsive airframe.
However, they are directly connected to these disadvantages: Stall will happen inboard first, which will result in a pitch-up.
If the tail cannot compensate, the stall is unrecoverable.
The aeroelastic effects will encourage flutter.
If flutter speed is found to be too low, the wing needs to be made stiffer, resulting in a weight increase.
These two disadvantages easily outweigh the advantages for most designs.
The first forward swept design, the Junkers 287 , needed a massive, wrought wing carry-through for stiffness.
Handling was great, but performance overall was poor.
Ju-287 with tufted wings and rockets below the wing engines.
The tripod ahead of the vertical holds a faired camera for in-flight observation of the tufts.
(picture source )
If combined with a T-tail, the aft position of the root of a forward swept wing will ensure that the tail is in the wake of the wing root at a high angle of attack, which can result in a deep stall .
The best forward-swept wings can be found in nature: Birds will sweep their wing tips forward if they want to pitch up and create lots of lift.
Especially for short landings the forward-swept wing is ideal when the legs are on the ground at the time of wing stall.
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For a text book overview: http://www.desktop.aero/appliedaero/potential3d/FSW.html Note that the Hansa Jet actually was a production airliner using the forward swept wing, and 2 glider types that i regulalrly flew also had forward swept wings (for visibility reasons: Schleicher Ka7 and Ka13). R.
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Also Zlin 142 has forward swept wing.
Not for aerodynamic reasons, but to place the pilots at the center of gravity and the spar behind them.
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Blanik L-13 and Scheibe Bergfalke sailplanes had swept-forward wings, it seems it helps in adding freedom in setting center of gravity respect to the pressure center and pilots placement.
Advantages of forward swept wings may be mainly in the low speed range, takeoff and landing.
I'm not too sure about Ju-287 being a failure, there are reports it reached 1'000 km/ h in the Soviet hands, good for a bomber, but developing the concept was discontinued, they may have found some heavy problems.
A tailless US airplane of the late 40s used this arrangement, NASA/ NACA, Cranfield repository has documents about it, also YouTube videos exist
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What are the advantages of a forward-swept wing vs. a rearward swept one? Image credit: NASA  Image credit:  Paul Maritz  / Wikimedia
What are the advantages of a forward-swept wing over a rearward-swept wing?
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<aircraft-design><wing>
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A froward-swept wing gives increased maneuverability and the wing spar carry-through can be placed aft of the cabin, so cabin height can be increased. The mentioned advantages of a forward-swept wing architecture are freedom in setting the center of gravity and the increased cabin height.
Some advantages  are stated, regarding the application of the building architectures.