Self-similarity is defined in a wide sense as the property of somesystems to be, either exactly or statistically, similar to a part ofthemselves. This property is found in certain geometric objects thatare intrinsically embedded in metric spaces, so that distance in themetric space gives a natural standard of measurement to uncoversimilar patterns at different observation scales. In complex networks,the definition of self-similarity is not obvious since many networksare not explicitly embedded in any physical geometry. In the absenceof a natural geometry, the main problem in the definition ofself-similarity stems from the fact that there is, a priori, no way todecide what is the part of the system that should be compared to (andlook alike) the whole. In this sense, self-similarity is not anintrinsic property of the system but it is directly related to thespecific procedure to identify the appropriate subsystem. In thistalk, I will explain how to define self-similarity in ensembles ofnetworks and multiplexes. Self-similarity has important implicationsin the global structure of networks and, in particular, in theirvulnerability to failures of their constituents. For instance,self-similarity alone —independently of the divergence of the secondmoment of the degree distribution— explains the absence of apercolation threshold in random scale-free networks. In the case ofself-similar multiplexes, we show that interlayer degree correlationscan change completely their global connectivity properties, so thatthey can even recover a zero percolation threshold and a continuoustransition in the thermodynamic limit, qualitatively exhibiting thusthe ordinary percolation properties of noninteracting networks.