If something can’t be detected, can we actually say it exists? This was the conundrum facing Wolfgang Pauli when he proposed that a neutral particle (i.e. the neutrino) is emitted in the beta decay of a nucleus. While it solved some profound mysteries, it was soon suggested that such a particle would be impossible to detect. Nearly a century later, we not only have detected neutrinos, but understand that they come in  three species, have antimatter counterparts, and that they play a crucial role as fundamental building blocks of the Universe and in determining its structure. I will discuss how neutrino oscillations, the transmutation of the neutrino among its three species, have uncovered some of its fundamental properties (such as the fact that it has a tiny but non-zero mass). Befitting its history, these studies have answered some questions about neutrinos but the answers are bizarre and raise new questions. In the near future, we hope to understand the relationship between the neutrino and antineutrino, which may provide critical clues in the continuing paradox of how the universe came to be dominated by matter.