One of the remarkable mathematical inventions of the 20th century is that of Fuzzy sets by Lotfi.A.Zadeh in 1965. His aim was to develop a mathematical theory to deal with uncertainty and imprecision. Fuzzy logic and the theory of fuzzy sets have been applied widely in areas like information theory, pattern recognition, clustering, expert systems, database theory, control theory, robotics, networks and nanotechnology. A.Rosenfeld introduced fuzzy graphs in 1975 to deal with relations involving uncertainty. In this book we present basic concepts in fuzzy graph connectivity, which plays a remarkable role in information networks and quality based clustering. This book consists of seven chapters. The first chapter includes motivation and basic results. Arc analysis of fuzzy graph structures, cycles in fuzzy graphs, blocks in fuzzy graphs, cycle connectivity of fuzzy graphs are discussed in the subsequent chapters. We believe that this book will help students, researchers and faculty of different institutes around the world to do fruitful research in fuzzy graph theory and related areas.