In this talk we will consider some notions of `robustness' of graph/hypergraph properties. We will survey some existing results and will try to emphasize the following new result (joint with Adva Mond and Kaarel Haenni): The binomial random digraph $D_{n,p}$
typically contains the minimum between the minimum out- and in-degrees many edge-disjoint Hamilton cycles, given that $p\geq \log^C n/n$, which is optimal up to log factors.