We present and explore a model of stateless and self-stabilizing distributed computation, inspired by real-world applications such as routing on today's Internet. Processors in our model do not have an internal state, but rather interact by repeatedly mapping incoming messages ("labels") to outgoing messages and output values. While seemingly too restrictive to be of interest, stateless computation encompasses both classical game-theoretic notions of strategic interaction and a broad range of practical applications (e.g., Internet protocols, circuits, diffusion of technologies in social networks). Our main technical contribution is a general impossibility result for stateless self-stabilization in our model, showing that even modest asynchrony (with wait times that are linear in the number of processors) can prevent a stateless protocol from reaching a stable global configuration. Furthermore, we present hardness results for verifying stateless self-stabilization. We also address several aspects of the computational power of stateless protocols. Most significantly, we show that short messages (of length that is logarithmic in the number of processors) yield substantial computational power, even on very poorly connected topologies.