In this talk, I will first give a bit of general context for higher categories and then a broad overview of some recent developments in higher category theory, in particular concerning higher categories nicknamed '$(\infty,2)$-categories' and 'double $\infty$-categories'.
Links:
[1] http://www.mpim-bonn.mpg.de/de/taxonomy/term/39
[2] http://www.mpim-bonn.mpg.de/de/node/3444
[3] http://www.mpim-bonn.mpg.de/de/node/3207