Tachyons do not exist in quantum theory. There are three reasons: their energy spectrum is unbounded from below, the vacuum state is unstable and not Lorentz-covariant, and the commutation rules are wrong. We critically examine the reasoning behind those arguments and find that the apparent obstacles for a quantum theory of tachyons are neatly removed if we extend the Hilbert space of tachyonic states in a two-state interpretation of quantum mechanics put forth by Aharonov et al. sixty years ago. Physical consequences of this new scheme include description of phase transitions in the early Universe.