It is well-known that the heat equation is not exactly controllable from the boundary because of the smoothing effect of its kernel. One can then ask which states are reachable at a given time. In this talk, we will give a definitive description of the reachable set of the one-dimensional heat equation on a segment as the Bergman space of a certain square. This result involves complex and harmonic analysis tools as a separation of singularities theorem for the Bergman space. Finally, we will give an overview of the open problems related to this result. This talk is based on joint works with Andreas Hartmann.