Summary: A graph \(X\) is called a weakly vertex-transitive graph (or weakly vertex-symmetric graph) if the endomorphism monoid End \(X\) of \(X\) acts transitively on the vertex set \(V(X)\). The weakly edge-transitive (or weakly edge-symmetric) graph is similarly defined. In this paper, some properties of graphs with weak symmetries are obtained and their endomorphism monoids are discussed.