  The structure of many real-world systems is best captured by networks
consisting of several interaction layers. Understanding how a multi-layered
structure of connections affects the synchronization properties of dynamical
systems evolving on top of it is a highly relevant endeavour in mathematics and
physics, and has potential applications to several societally relevant topics,
such as power grids engineering and neural dynamics. We propose a general
framework to assess stability of the synchronized state in networks with
multiple interaction layers, deriving a necessary condition that generalizes
the Master Stability Function approach. We validate our method applying it to a
network of R\"ossler oscillators with a double layer of interactions, and show
that highly rich phenomenology emerges. This includes cases where the stability
of synchronization can be induced even if both layers would have individually
induced unstable synchrony, an effect genuinely due to the true multi-layer
structure of the interactions amongst the units in the network.
