We analyze tail asymptotics of a two-node tandem queue with spectrally-positive Lévy input. A first focus lies in the tail probabilities of the type ℙ(Q1>αx,Q2>(1−α)x), for α∈(0,1) and x large, and Qi denoting the steady-state workload in the ith queue. In case of light-tailed input, our analysis heavily uses the joint Laplace transform of the stationary buffer contents of the first and second queue; the logarithmic asymptotics can be expressed as the solution to a convex programming problem. In case of heavy-tailed input we rely on sample-path methods to derive the exact asymptotics. Then we specialize in the tail asymptotics of the downstream queue, again in case of both light-tailed and heavy-tailed Lévy inputs. It is also indicated how the results can be extended to tandem queues with more than two nodes.