The distance of a vertex in a connected graph is the sum of distances from the vertex to all other vertices. The distance of a graph is half of the sum of distances of its vertices. The concept of the distance of a graph has strong motivation from chemistry, where the same concept is known as Wiener index. The paper surveys a number of results on distances of trees and distances of vertices of trees. Some results in the paper are new, e.g. the bounds on sums of reciprocals of degrees in trees.