With its use of the powerful technology of computer, the computational theory of mind or computationalism, which regards minds as computational systems, has been widely hailed as the most promising theory that will carry out the project of explaining the workings of the mind in purely scientific terms. While it continues to serve as the primary framework for scientifically inclined theorizing and investigations about the nature of minds, especially in the area of cognitive science, it, however, continues to face strong objections from its critics. And with the growing complexity and sophistication of the arguments used to promote and reject the theory, the debate has become intractable. It has become quite difficult to assess which side of the dispute is gaining the upper hand. Such difficulty may be due to a variety of reasons. In this essay, I critically examine two of such reasons. The first concerns the ambiguity of the theory’s intended scope of application: whether it is limited to the mind’s cognitive features only or it also includes the mind’s phenomenal features. The second concerns the vagueness of how the so-called computer modelling of human cognitive processes is able to duplicate such processes. Accordingly, if insufficiently addressed, they remain as two roadblocks to the entire project of computationalism.