what is the image of the heat operator? to put it the other way around, what is the domain of the backward heat operator? that is, for which initial conditions is it possible to solve the backward heat equation (say, for some fixed time t)? clearly, the initial conditions must be extremely regular, since the forward heat equation is highly smoothing. in the case of euclidean space $r^{n}$, there is a simple answer: the initial conditions must admit an analytic continuation to $c^n$ with a certain (t-dependent) growth in the imaginary directions. for compact symmetric spaces (e.g. a sphere), there is a very similar answer. for noncompact symmetric spaces (e.g. hyperbolic space), the situation is much more complicated and we are just beginning to understand what is going on. i will describe some old and some new results in this area, which can be expressed in terms of a ‘‘eneralized segal--bargmann transform’’