Symmetry of graphs has been extensively studied over the past 50 years by using automorphisms of graphs and group theory which have played an important role in graph, and promising and interesting results have been obtained [N. Biggs, Algebraic Graph Theory, Cambridge University Press, Cambridge, 1993, C. Godsil, C. Royle, Algebraic graph theory, Springer-Verlag, London, 2001, G. Hahn, G. Sabidussi, Graph Symmetry (algebraic methods and application), Kluwer Academic Publishers, Dordrecht, 1997]. We introduced generalized symmetry of graphs and investigated it by using endomorphisms of graphs and semigroup theory. In this paper, I will survey some results we have achieved in recent years. The paper consists of the following sections.

1. Introduction

2. End-regular graphs

3. Weakly transitive graphs

4. Cayley graphs of semigroups

5. Unretractive graphs

6. Graphs and their endomorphism monoids.