Theories that the full moon influences human behavior are strongly rooted in superstition and many can be traced back to the middle ages in Europe. Increased aggression leads to more violent crimes, more accidents and more suicides all during full moons. The modern word "lunacy" which means insanity is associated with the Moon ("luna" is Latin for moon). In reality though, scientific studies have been inconclusive and inconsistent at best in finding correlations between the full moon and changing behavior. Most studies have found no evidence at all.
The Moon, magic and mythology
The Moon was an important part of many ancient religions and rituals. Gods and goddesses associated with the Moon were ones shrouded in mystery and often associated with magic (Hecate from Greece) and with death (Thoth from Egypt). From the Native Americans to the Ancient Chinese and everyone in between, there were moon-related myths because peoples' lives revolved around the seasons and the skies. These ancient people also realized the effect that the Moon had on the tides.
All of these traces of ancient religions trickled through society. During the middle ages, Europe was wrought with illness from the Black Plague and superstition was commonplace, while education was not. The Moon became associated with witchcraft and various misdeeds, along with black cats and other innocuous things. These superstitions have clung to us even today in our modern and scientifically advanced society.
Why do we still believe the Moon influences behavior?
The unfounded theories about the Lunar Effect or Transylvania Effect still abound because of a psychological tendency of humans to take note of coincidences. In the back of our minds, we are all familiar with the idea of the Moon playing a role in behavior and "causing" odd happenings. We then tend to remember when "odd" events happen during a full moon, which gives them an illusion of being more odd. The same event can happen during another phase of the moon and we will not think twice about it.
Our expectations influence our perceptions, just as with the so-called "Blue Car Effect." If you buy a blue car, you begin to notice blue cars everywhere. If you believe the full moon affects behavior, you will notice behavior during the full moon. This phenomena can also severely skew the results of some studies if the researchers are not careful to account for it. It could be the reason of the few studies that have found correlations between the full moon and behavior-related occurrences.