  Graphene has emerged as the foremost material for future two-dimensional
spintronics due to its tuneable electronic properties. In graphene, spin
information can be transported over long distances and, in principle, be
manipulated by using magnetic correlations or large spin-orbit coupling (SOC)
induced by proximity effects. In particular, a dramatic SOC enhancement has
been predicted when interfacing graphene with a semiconducting transition metal
dechalcogenide, such as tungsten disulphide (WS$_2$). Signatures of such an
enhancement have recently been reported but the nature of the spin relaxation
in these systems remains unknown. Here, we unambiguously demonstrate
anisotropic spin dynamics in bilayer heterostructures comprising graphene and
WS$_2$. By using out-of-plane spin precession, we show that the spin lifetime
is largest when the spins point out of the graphene plane. Moreover, we observe
that the spin lifetime varies over one order of magnitude depending on the spin
orientation, indicating that the strong spin-valley coupling in WS$_2$ is
imprinted in the bilayer and felt by the propagating spins. These findings
provide a rich platform to explore coupled spin-valley phenomena and offer
novel spin manipulation strategies based on spin relaxation anisotropy in
two-dimensional materials.
