Gossip protocols, also known as epidemic dissemination schemes, are becoming increasingly popular in distributed systems. Yet, it has remained a partially open question to determine how robust such protocols can be. We consider a natural extension of the random phone-call model by Karp et al. [21], in which we analyze two different notions of robustness: the ability to tolerate adaptive failures, and the ability to tolerate oblivious failures. For adaptive failures, we present a new gossip protocol, TrickleGossip, which achieves near-optimal O(n log3 n) message complexity; to the best of our knowledge, this is the first epidemic-style protocol that can tolerate adaptive failures. We also show a direct relation between resilience and message complexity, demonstrating that gossip protocols which tolerate a large number of adaptive failures need to use a super-linear number of messages with high probability. For oblivious failures, we present a new gossip protocol, CoordinatedGossip, that...