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The space of null geodesics of a spacetime (a Lorentzian manifold with a choice of future) sometimes has the structure of a smooth manifold. When this is the case, it comes equipped with a canonical contact structure. I will introduce the theory for a countable number of metrics on the product $S^2\times S^1$. Motivated by these examples, I will comment on how Engel geometry can be used to describe the manifold of null geodesics, by considering the Cartan deprolongation of the Lorentz prolongation of the spacetime. This allows us to characterize the 3-contact manifolds which are spaces of null geodesics, and to retrieve the spacetime they come from. This is joint work with R. Rubio.