  Many real complex systems cannot be represented by a single network, but due
to multiple sub-systems and types of interactions, must be represented as a
multiplex network. This is a set of nodes which exist in several layers, with
each layer having its own kind of edges, represented by different colours. An
important fundamental structural feature of networks is their resilience to
damage, the percolation transition. Generalisation of these concepts to
multiplex networks requires careful definition of what we mean by connected
clusters. We consider two different definitions. One, a rigorous generalisation
of the single-layer definition leads to a strong non-local rule, and results in
a dramatic change in the response of the system to damage. The giant component
collapses discontinuously in a hybrid transition characterised by avalanches of
diverging mean size. We also consider another definition, which imposes weaker
conditions on percolation and allows local calculation, and also leads to
different sized giant components depending on whether we consider an activation
or pruning process. This 'weak' process exhibits both continuous and
discontinuous transitions.
