The most famous and mysterious plane disappearance of all time was that of Amelia Earhart. She was the first woman to fly solo across the Atlantic Ocean, which she did at the age of 30. In 1937, during her journey to circumvent the globe, she went missing near Howland Island in the Pacific Ocean. Even though the most likely story is a crash, there are many theories as to what happened to Earhart -- including wild ideas like alien abduction or that she was captured by the Japanese.
A multi-million dollar search was performed but no signs of Earhart or her plane were ever found. An explorer finally claimed to have found a piece of her plane in 2014, but has limited evidence to support his theory.