Robustness of graph properties

A typical result in graph theory says that a graph $G$, satisfying certain conditions, has some property $\cal P$. Once such a theorem is established, it is natural to ask how strongly $G$ satisfies $\cal P$. Can one strengthen the result by showing that $G$ possesses $\cal P$ in a robust way? What measures of robustness can one utilize? In this survey, we discuss various measures that can be used to study robustness of graph properties, illustrating them with examples.