Fixed point theorems have been used extensively in solving Ulam-Hyers stability problems. In this chapter we will focus on Ulam-Hyers stability of different types of functional equations in normed spaces, their modifications and generalizations like \(\beta \) -normed spaces, p-normed spaces, non-Archimedean normed space, fuzzy and random normed spaces. We collect the new results on this topic with following fixed point theorems that were used in their acquiring. Wide range of approaches to the proof of some well-known stability results is also presented from the angle of Fixed Point Theory.