In this article we discussed prominence of Fuzzy Eulerian and Fuzzy Hamiltonian graphs. Fuzzy logic is introduced to study the uncertainty of the event. In Fuzzy set theory we assign a membership value to each element of the set which ranges from 0 to 1. The earnest efforts of the researchers are perceivable in the relevant establishment of the subject integrating coherent practicality and reality. Fuzzy graphs found an escalating number of applications in day to day life system where the information intrinsic in the system varies with different levels of accuracy. In this article we initiated the model of fuzzy Euler graphs (FEG) and also fuzzy Hamiltonian graphs (FHG). We explored about fuzzy walk, fuzzy path, fuzzy bridge, fuzzy cut node, fuzzy tree, fuzzy blocks, fuzzy Eulerian circuit and fuzzy Hamiltonian cycle. Here we studied some applications of Fuzzy Eulerian graphs and fuzzy Hamiltonian graphs in real life.