In 1928 the British physicist Paul Dirac wrote down an equation for the quantum theory of the electron. Dirac’s equation had a strange property, it had two solutions just like the equation a2 = 4 (which has solutions a=+2 and a=-2).
One of these solutions corresponded to the motion of an electron. The other solution to a particle with apparently negative energy. Dirac theorised that this second solution must correspond to a positive energy antiparticle, the positron.
Dirac’s positron was observed by an American physicist Carl Anderson four years later in 1932. Photos from Anderson’s cloud chamber experiment showed unexpected particle tracks. The particle appeared to behave a lot like an electron but had the wrong electric charge for an electron (curving in the wrong direction in the experiments magnetic field). Anderson's observation confirmed Dirac's prediction and antimatter was born.
Dirac shared the 1933 Nobel Prize with the Austrian physicist Erwin Schroedinger for his contributions to quantum theory. Anderson was awarded a Nobel Prize for discovering antimatter in 1936.