  In many systems consisting of interacting subsystems, the complex
interactions between elements can be represented using multilayer networks.
However percolation, key to understanding connectivity and robustness, is not
trivially generalised to multiple layers. We describe a generalisation of
percolation to multilayer networks: weak multiplex percolation. A node belongs
to a connected component if at least one of its neighbours in each layer is in
this component. We fully describe the critical phenomena of this process. In
particular, in two layers, with finite second moments of the degree
distributions, an unusual continuous transition with quadratic growth above the
threshold. When the second moments diverge, the singularity is determined by
the asymptotics of the degree distributions, creating a rich set of critical
behaviours. In three or more layers we find a discontinuous hybrid transition
which persists even in highly heterogeneous degree distributions, becoming
continuous only when the powerlaw exponent reaches $1+ 1/(M-1)$ for $M$ layers.
