From what I read, "pusher" propeller-driven aircraft have certain theoretical aerodynamic advantages over conventional "tractor" aircraft (where the propeller is positioned in front of the engine).
Though it's not obvious from looking at a photo or a prop-driven airplane in flight, a forward-facing propeller whips a lot of air into a spiral pattern that twists around even a highly streamlined fuselage (if it's a single-engine aircraft) and disrupts airflow over the wings almost no matter where the propeller is positioned.
The solution to all this nasty propeller-induced turbulence? -- place the propeller behind the engine so that it pushes rather than pulls, leaving the churned air behind the plane rather than engulfing important parts of it.
Even though the convention was the tractor arrangement, a number of early airplanes were pusher types including the Wright Brothers' 1903 flyer and some Great War combat planes. From time to time aircraft with tractor and pusher propellers were built. But designers quickly and nearly universally preferred the tractor arrangement.
The siren song of the pusher was heard once again in the late 1930s and early 40s for military planes. This post deals with some designs that reached flying prototype status, but only two of which entered real military service.
Despite the potential advantages mentioned above, the pusher configuration had its limitations. One was that it was often difficult to provide sufficient cooling for the motors. This is obvious for air-cooled engines requiring a large blast of air passing around the cylinders, but apparently was even the case for water-cooled engines requiring radiators.
Another problem was propeller clearance on takeoff and landing. There was a risk the the props might dig in to the ground when the aircraft was assuming a nose-up attitude. Tractor aircraft had much less to worry about in this regard.
But perhaps the greatest problem was what to do when the pilot had to bail out of a single-engine plane; without special steps taken, he would be chopped up as he passed through the propeller arc. Solutions included feathering the prop or detaching it before bail-out. Another solution was the ejection seat common on jet fighters but something that didn't emerge until towards the end of World War 2.
It is possible that the ejection seat and contra-rotating props of comparatively small diameter might have let to successful fighter designs by the late 1940s. But the advent of jet propulsion along with diminishing returns to increased piston engine power ended propeller-driven fighter planes regardless of whether they were tractor or pusher. The only real pusher success was a huge bomber, as noted below.
YFM-1 was a pre-war concept for a plane that could serve either as a bomber escort or an interceptor. It had a five-man crew including a 37-mm cannon operator in the front of each engine nacelle (and subject to being decimated on bail-out). The planes were too heavy for the power available and the pusher system created pitching problems. There were even more problems -- see the link for an enumeration -- so that even though a test batch had been ordered, the Airacuda never entered full-scale production.
The XP-54, along with the XP-55 and XP-56 were results of the Army's attempt to unleash airplane designers to create wild and crazy stuff. For some reason, all three participants in the program opted for the pusher layout. Every project resulted in disappointment, one important factor being that the engines planned initially were technical failures and replacement motors lacked enough power for performance superior to existing production models such as the P-51 Mustang. The XP-54 was the most conventional design, featuring a twin-boom arrangement to support the tail.
The XP-55 was more radical, having a "canard" layout where instead of tail-mounted horizontal stabilizers, they are placed near the nose.
Most radical of all was the XP-56 which had no stabilizers at all, being close to a flying wing design with a fat fuselage. Note that the XP-56 had contra-rotating props with a smaller diameter of arc than a single-prop version might have. This would lessen the risk of prop dig-in on takeoff rotation or landing.
This Swedish fighter actually reached production and saw service in the late 1940s and early 50s. Interestingly, it was adapted to jet power and that version also was produced.
Two Army pusher configuration "intercontinental" range bomber designs reached prototype and test-batch status in the late 1940s. The XB- and YB-35s are manifestations of Jack Northrop's fascination with the flying wing concept. Like the J21, the YB-35 was adapted to jet power as the YB-49, but production contracts went to the plane below.
Several hundred B-36 bombers were built and served in the U.S. Air Force's Strategic Air Command during the 1950s. It featured a pusher layout soon augmented with four jet engines mounted in pods below the wings for greater target-dash speed. Although the B-36 had its problems, it stands as the most successful military pusher aircraft since the Great War. The B-36 was replaced by the eight-jet B-52 which has remained in service for more than half a century.