  Let $(X^{m+1}, g)$ be a globally hyperbolic spacetime with Cauchy surface
diffeomorphic to an open subset of $\mathbb R^m$. The Legendrian Low conjecture
formulated by Nat\'ario and Tod says that two events $x,y\in\ss$ are causally
related if and only if the Legendrian link of spheres $\mathfrak S_x, \mathfrak
S_y$ whose points are light geodesics passing through $x$ and $y$ is
non-trivial in the contact manifold of all light geodesics in $X$. The Low
conjecture says that for $m=2$ the events $x,y$ are causally related if and
only if $\mathfrak S_x, \mathfrak S_y$ is non-trivial as a topological link. We
prove the Low and the Legendrian Low conjectures. We also show that similar
statements hold for any globally hyperbolic $(X^{m+1}, g)$ such that a cover of
its Cauchy surface is diffeomorphic to an open domain in $\mathbb R^m.$
