A shell-and-tube heat exchanger for handling an easily polymerizable substance is disclosed, which is characterized by having no part for stagnation of said process fluid on the surfaces for contact with the process fluid. By eliminating undulations from the inner surfaces of the heat transfer tubes, it is made possible to prevent the easily polymerizable substance from polymerizing and, at the same time, prevent adhesion of a polymer to the surfaces. Thus, the efficiency of heat exchange is enhanced and the protracted continuous operation is realized.