What are the advantages of a forward-swept wing over a rearward-swept wing?
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.