The Chinese are attributed with the invention in AD 105 of what we know today as paper. A good bet is that the first flying paper object soon followed as a screwed up paper ball flew toward a trash can. Recognizable paper airplanes appeared much later, with sightings coinciding with the first powered flight by the Wright brothers at the beginning of the 20th Century.
Many an aeronautical engineer started out as a paper airplane designer and manufacturer. In fact, who hasn't built a paper airplane and experienced the joy and frustration of such a simple toy? As with many such endeavors the force of competition is never far away and so it goes for paper airplanes. The current world record holder (as of 2007) for the longest duration paper airplane flight (at 27.6 seconds) is Ken Blackburn. Maybe it's no coincidence that Ken is also an aeronautical engineer and brings a deep understanding of aerodynamics to his hobby, or more accurately – his obsession.
Maximizing Flight Duration
The design of Ken's world record holding paper airplane is remarkably simple with a low-aspect-ratio (short and stubby) rectangular wing. Gliders typically favor high-aspect-ratio (long and skinny) wings that maximize the glide time (by maximizing their lift/drag ratio) before it's time to land. Ken's choice of a low-aspect-ratio seems to contradict accepted wisdom on airplane design until you see how Ken launched his airplane – he throws it directly upwards at 60mph. A high-aspect-ratio wing, made of paper could not withstand the launch forces, so Ken goes for a robust low-aspect-ratio wing. The descent speed is higher, but so too is the height his airplane reaches prior to its descent. This compromise in aerodynamic performance and radical launch seems to be a near-optimal solution to maximizing the flight duration.
Maximizing Flight Distance
If the measure of performance is switched to the maximum distance flown from launch, then a high-aspect-ratio wing should find favor over a low-aspect-ratio wing. The world record for the longest distance flown by a paper airplane is 193 feet (58.82m), held by Tony Fletch of Wisconsin, USA – though I can't find a picture of his design to determine the characteristics of his airplane.
Airflow simulation using Computational Fluid Dynamics (CFD) could offer some insight into flight performance of paper airplanes, but the simulation would also have to incorporate structural interactions due to the flexibility of paper in flight. This form of simulation known as Fluid-Structure Interaction (FSI) is complex and time consuming. Another problem with paper airplanes is the variability of manufacture at such a small scale can have a significant effect on flight characteristics.
There's a chance that the progression of CPU performance according to Moore's Law will one day make the complex analysis of such a simple toy trivial, but until that day, keep on folding and flying. How marvelous – going from concept to first flight in a matter of seconds.