We present path splicing, a new routing primitive that allows network paths to be constructed by combining multiple routing trees ("slices") to each destination over a single network topology. Path splicing allows traffic to switch trees at any hop en route to the destination. End systems can change the path on which traffic is forwarded by changing a small number of additional bits in the packet header. We evaluate path splicing for intradomain routing using slices generated from perturbed link weights and find that splicing achieves reliability that approaches the best possible using a small number of slices, for only a small increase in latency and no adverse effects on traffic in the network. In the case of interdomain routing, where splicing derives multiple trees from edges in alternate backup routes, path splicing achieves near-optimal reliability and can provide significant benefits even when only a fraction of ASes deploy it. We also describe several other applications of path splicing, as well as various possible deployment paths.