Hardy's celebrated original inequality from the 1920's was formulated as a discrete inequality on the natural numbers. Since then it became most relevant in its various continuum versions. For example in mathematical physics it serves as a quantitative version of Heisenberg's uncertainty principle and it is a most powerful tool in partial differential equation. We return to the discrete setting of graphs and discuss how Hardy inequalities can be obtained and explain how optimality can be shown. We illustrate this by various examples. This also leads us to a disparity to the continuum setting namely that Hardy's original inequality can be improved in the discrete setting. Finally, we explain how Hardy inequalities can be used to derive Rellich inequalities control the growth of solutions. (This includes joint work with F. Fischer, M. Lemm, M. Nietschmann, F. Pogorzelski and Y. Pinchover)