Serial murder is undoubtedly one of the most terrifying and fascinating contemporary crimes, both in academic research and also in everyday popular culture. But what exactly is the phenomenon of serial murder? How do we define and evaluate it? In this essay, I review the nature of serial murder and then examine the phenomenon within the context of current R.O.C. criminal law. I raise several questions: Is this type of murder a consecutive offence? In what ways does this crime relate to the concept of insanity or mental disorder? And how should we deal with serial murder-capital or life punishment? Capital punishment seems an easy option, but is not in accordance with international criminal justice policy trends. This essay also deals with issue of criminal response to serial murder.