Let X be a normed space, G a nonempty bounded subset of X and {x n } a bounded sequence in X. In this article, we introduce and discuss the concept of asymptotic farthest points of {x n } in G, which is a new definition in abstract approximation theory. Then, by applying the topics of functional analysis, we investigate the relation between this new concept and the concepts of extreme points and convexity. In particular, one of the main purposes of this paper is to study conditions under which the existence (uniqueness) of asymptotic farthest point of {x n } in G is equivalent to the existence (uniqueness) of asymptotic farthest point of {x n } in ext(G) or co(G).