The attainable region (AR) is a region in a performance space in which all physically realizable reactor network designs must exist. ARs have been used since the 1960s for solving reactor network synthesis problems. The benefits of these methods are that the feasibility of a performance target can be assessed prior to running a synthesis routine, the solutions they give are guaranteed to be physically realizable, and a design can be made robust to uncertainties in feed and performance targets by assessing whether a solution and the range of its possible values lie within the AR, just to name a few. In this article, the theory of attainable regions is extended to bring these benefits to the heat exchanger network (HEN) synthesis problem. Basic properties of the HEN-AR are proven and a synthesis method using the AR is presented with examples.