This article discusses the theory and practice of the insight approach to mediation; an approach that applies Lonergan's philosophy of cognition to the field of conflict resolution. The authors focus on a specific type of learning conversation, known as deepening, that is important in the practice of insight mediation. To help us understand deepening conversations, they begin with an overview of the theoretical foundations from Lonergan on which the practice of deepening is based and then go on to describe key aspects of insight mediation. In the latter part of the article, and to illustrate the application of the theory to the practice of mediation, they take us through a simulated dialogue that involves a conflict between a father and daughter over the daughter's pending marriage.