We introduce a new method, called Priority Encoding Transmission, for sending messages over lossy packet-based networks. When a message is to be transmitted, the user speci es a priority value for each part of the message. Based on the priorities, the system encodes the message into packets for transmission and sends them to (possibly multiple) receivers. The priority value of each part of the message determines the fraction of encoding packets su cient to recover that part. Thus, even if some of the encoding packets are lost enroute, each receiver is still able to recover the parts of the message for which a su cient fraction of the encoding packets are received. International Computer Science Institute, Berkeley, California. Research supported in part by National Science Foundation operating grant NCR-941610 yComputer Science Department, Swiss Federal Institute of Technology, Zurich, Switzerland. Research done while a postdoc at the International Computer Science Institute, Berkeley...