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The Hamiltonian H specifies the energy levels and the time evolution of a quantum theory. It is an axiom of quantum mechanics that H be Hermitian. The Hermiticity of H guarantees that the energy spectrum is real and that the time evolution is unitary (probability preserving). In this talk we investigate an alternative formulation of quantum mechanics in which the mathematical requirement of Hermiticity is replaced by the more physically transparent condition of space-time reflection (𝒫𝒯) symmetry. We show that if the 𝒫𝒯 symmetry of a Hamiltonian H is not broken, then the spectrum of H is real. Examples of 𝒫𝒯-symmetric non-Hermitian Hamiltonians are H = p2 + ix3 and H = p2 − x4. The crucial question is whether 𝒫𝒯-symmetric Hamiltonians specify physically acceptable quantum theories in which the norms of states are positive and the time evolution is unitary. The answer is that a Hamiltonian that has an unbroken 𝒫𝒯 symmetry also possesses a physical symmetry that we call C. Using C, we show how to construct an inner product whose associated norm is positive definite. The result is a new class of fully consistent complex quantum theories. Observables exhibit 𝒞𝒫𝒯 symmetry, probabilities are positive, and the dynamics is governed by unitary time evolution.