  In recent years there has been much progress in graph theory on questions of
the following type. What is the threshold for a certain large substructure to
appear in a random graph? When does a random graph contain all structures from
a given family? And when does it contain them so robustly that even an
adversary who is allowed to perturb the graph cannot destroy all of them? I
will survey this progress, and highlight the vital role played by some newly
developed methods, such as the sparse regularity method, the absorbing method,
and the container method. I will also mention many open questions that remain
in this area.
