A typical result in graph/hypergraph theory has the following structure: Every G satisfying certain conditions must have some target property P. For example, a classical theorem by Dirac asserts that every graph on n vertices and with minimum degree at least n/2 must contain a hamiltonian cycle (that is, a cycle that passes through every vertex).

After establishing such a theorem, it is natural to ask how ``robust'' is G with respect to this property P. In this talk we discuss some possible measures of ``robustness'' and illustrate them with many examples.