As defined in Chap. 2, a fuzzy graph without fuzzy cutvertices is called a block (nonseparable). Rosenfeld introduced this concept in 1975. In contrast to the classical concept of blocks in graphs, the study of blocks in fuzzy graphs is challenging due to the complexity of cutvertices. Note that cutvertices of a fuzzy graph are those vertices which reduce the strength of connectedness between some pair of vertices on its removal from the fuzzy graph rather than the total disconnection of the fuzzy graph. In this chapter, we concentrate on blocks of fuzzy graphs. This work is from Anjali and Mathew, J Fuzzy Math, 23(4), 907–916 (2015), , Anjali and Mathew, J Intell Fuzzy Syst 28, 1659–1665 (2015), , Anjali and Mathew, Transitive blocks in fuzzy graphs (2017), .