This chapter introduces and collects some Liapunov functions useful in the application of the Direct Method. It emphasizes that the goal is merely to introduce possible Liapunov functions for some equations of general form. In this strategy a central role is played by the inequalities that one is able to obtain from the P.D.E. at hand. In order to be more precise, let U be a scalar function of a time like variable t – linked in some definite manner to the state vector. The chapter begins by recalling a very well known general estimate obtained from the first order inequality, that is, the so called Gronwall lemma and its generalization. In 1893 A.M.Liapunov – in order to establish conditions ensuring stability of solutions of O.D.E.s – introduced a method which is called the direct or second method.