The notions of centrality and distance-based consensus are important concerns in many areas such as social network theory and classification theory. The general set-up consists of a finite metric space X and a subset S of X. For x in X, let D(x,S) be a measure of 'remoteness' of x to S, and let L be the function where L(S) is the set of all points x in X for which D(x,S) is minimum. L is called the median function on X when D(x,S) is the sum of distances of x to all the points in S, L is called the mean function on X when D(x,S) is the sum of the squared distances, and L is called the center function on X when D(x,S) is the maximum of the distances of x to all the points in S. This talk will review recent results obtained toward characterizing the median, mean and center functions on metric spaces such as certain classes of graphs (symmetric networks) and spaces of various types of classifications on a fixed set of entities.