Familiar textbook quantum mechanics assumes a fixed background spacetime to define states on spacelike surfaces and their unitary evolution between them. Quantum theory has been generalized as our conceptions of space and time have evolved. But quantum mechanics needs to be generalized further for quantum gravity where spacetime geometry is fluctuating and without definite value. This talk will review a fully four-dimensional, sum-over-histories, generalized quantum mechanics of cosmological spacetime geometry. In this generalization, states of fields on spacelike surfaces and their unitary evolution are emergent properties appropriate when spacetime geometry behaves approximately classically. The principles of generalized quantum theory would allow for further generalization that would be necessary were spacetime not fundamental. Emergent spacetime phenomena are discussed in general and illustrated with the examples of the classical spacetime geometries with large spacelike surfaces that emerge from the `no-boundary' wave function of the universe. These must be Lorentzian with one, and only one, time direction. The question will be raised as to whether quantum mechanics itself is emergent.