In this paper, we introduce the problem of Continuous Gossip in which rumors are continually and dynamically injected throughout the network. Each rumor has a deadline, and the goal of a continuous gossip protocol is to ensure good "Quality of Delivery," i.e., to deliver every rumor to every process before the deadline expires. Thus, a trivial solution to the problem of Continuous Gossip is simply for every process to broadcast every rumor as soon as it is injected. Unfortunately, this solution has a high per-round message complexity. Complicating matters, we focus our attention on a highly dynamic network in which processes may continually crash and recover. In order to achieve good per-round message complexity in a dynamic network, processes need to continually form and re-form coalitions that cooperate to spread their rumors throughout the network. The key challenge for a Continuous Gossip protocol is the ongoing adaptation to the ever-changing set of active rumors and non-crashed process. In this work we show how to address this challenge; we develop randomized and deterministic protocols for Continuous Gossip and prove lower bounds on the per-round message-complexity, indicating that our protocols are close to optimal.