The vague question "What can machines do and what can they not do?" has generated much controversy in the past, and continues to be a source of confusion for many people. The increasingly pervasive role that computers play in our society makes it important that the public acquire some understanding of the potential and the limitations of machines. Over the past 40 years some rather precise questions have been formulated and partially answered, shedding light on the issue of what machines can and cannot do. An awareness and understanding of a few of these questions helps to avoid much confusion about the capabilities of machines, and to dispel some widespread myths. We have chosen to discuss the following three questions: 1) Can a machine think? 2) Can a machine reproduce itself? 3) Are there tasks we can prove no machine will ever be able to perform?