  This paper argues that non-self-adjoint operators can be observables. There
are only four ways for this to occur: non-self-adjoint observables can either
be normal operators, or be symmetric, or have a real spectrum, or have none of
these three properties. I explore each of these four classes of observables,
arguing that the class of normal operators provides an equivalent formulation
of quantum theory, whereas the other classes considerably extend it.
