In this talk we present a new generalization of the mathematical notion of distance. It is based on the abstraction of the codomain of the distance function. The resulting functions must satisfy a generalized triangular inequality, which depends only on the order structure of the valuation space, i.e., a monoid structure is not required. This type of function will be called i-Distance (i-metric, i-quasi-metric, etc). We show that they generate a topology in a very natural way based on open balls. This concept has been successfully applied in the field of Clustering Algorithms and Pattern Recognition. We show how the concept was applied in this field as well as some theoretical results.