Turns out that the influence in the tides is the only Lunar effect that is scientifically proved. It's gravitational force on the Earth is what makes the tides stronger in the Full and New Moons.
A curious information is that the fate of World War 2 was kind of determined by the Moon. That is because General Dwight Eisenhower only decided the date of landing in Normandy (June 6 1944) after consulting the Moon cale. Given that it would be a clear night, with low tides, it was ideal. The date was initially decided for June 5, but a strong storm made him change his mind... And to change the course of war.