At the age of 23, Paul Dirac developed in October 1925 his own version of quantum mechanics sometimes known as q-number algebra. He soon emerged as a pioneer of the field with fundamental contributions to quantum statistics, radiation theory, and more. Dissatisfied with the relativistic Klein-Gordon quantum equation, in the fall of 1927 he looked for a more satisfactory equation that described the electron in agreement with Einstein’s special theory of relativity. The result, published in early 1927, was the famous Dirac equation which explained the electron’s spin and also hydrogen’s fine-structure spectrum. To make physical sense of the new theory, he first interpreted the electron’s antiparticle as a proton, an idea he liked very much because it promised a reduction of all matter in terms of a single fundamental entity, the electron. However, in a remarkable paper of 1931 (which dealt principally with magnetic monopoles) he replaced the proton with a hypothetical positive electron. Two years later, when awarded the Nobel Prize, he speculated about stars made of antimatter consisting of negative protons (antiprotons) and positive electrons (positrons). Half of all matter in the universe might be antimatter, he boldly suggested. Focusing on the period 1927-1933, the talk will outline how Dirac created the equation named after him and subsequently formulated the radical idea of antiparticles and antimatter.