Roughly speaking, a Hardy type inequality for a nonnegative operator P is a functional inequality P>=C w, involving a weight function w which has to be taken as “large” as possible and a constant C. One particular focus in the literature lies on finding the sharp constant C to a prescribed Hardy-weight w which is typically an inverse square weight. The thesis will start with a short survey on the discrete and continuous version of the classical Hardy inequality. Then it will focus on more recent developments in the context of Schrödinger operators on weighted graphs. Finally, the analysis will be specified on infinite (possibly homogeneous) trees and the relationship between Hardy weights and superharmonic functions will be investigated.