The inverse degree of a finite graph is defined as , where is the degree of vertex . We establish inequalities concerning the sum of the diameter and the inverse degree of a graph which for the most part are tight. We also find upper bounds on the diameter of a graph in terms of its inverse degree for several important classes of graphs. For these classes, our results improve bounds by Erdős et al. (1988) [5], and by Dankelmann et al. (2008) [4].