The Turing Test is one of the most disputed topics in Artificial Intelligence, Philosophy of Mind and Cognitive Science. It has been proposed 50 years ago, as a method to determine whether machines can think or not. It embodies important philosophical issues, as well as computational ones. Moreover, because of its characteristics, it requires interdisciplinary attention. The Turing Test posits that, to be granted intelligence, a computer should imitate human conversational behavior so well that it should be indistinguishable from a real human being. From this, it follows that conversation is a crucial concept in its study. Surprisingly, focusing on conversation in relation to the Turing Test has not been a prevailing approach in previous research. This thesis first provides a thorough and deep review of the 50 years of the Turing Test. Philosophical arguments, computational concerns, and repercussions in other disciplines are all discussed. Furthermore, this thesis studies the Turing Test as a special kind of conversation. In doing so, the relationship between existing theories of conversation and human-computer communication is explored. In particular, Grice's cooperative principle and conversational maxims are concentrated on. Viewing the Turing Test as conversation and computers as language users have significant effects on the way we look at Artificial Intelligence, and on communication in general.