We present a new approach to deal with default information based on the theory of belief functions. Our semantic structures, inspired by Adams' epsilon semantics, are epsilon-belief assignments, where mass values are either close to 0 or close to 1. In the first part of this paper, we show that these structures can be used to give a uniform semantics to several popular non-monotonic systems, including Kraus, Lehmann and Magidor's system P, Pearl's system Z, Brewka's preferred sub-theories, Geffner's conditional entailment, Pinkas' penalty logic, possibilistic logic, and the lexicographic approach. In the second part, we use epsilon-belief assignments to build a new system, called LCD, and we show that this system correctly addresses the well-known problems of specificity, irrelevance, blocking of inheritance, ambiguity, and redundancy.