In liturgical traditions, the Rite of Peace or Sign of Peace is performed just before the Eucharist (I have only experienced the Catholic and Episcopalian versions, so don't know how far this reaches into non-liturgical churches). The ostensible purpose is to bring a spirit of fellowship into the meeting before that fellowship is perfected in the Eucharist. In both of the versions I have experienced, after a short exhortation the celebrant asks the congregation to give a Sign of Peace, and basically you go in a circle and shake hands with everyone around you while saying "Peace be with you" or whatever you want.
There is a lot of talk in the bloggernacle about the relative merits of members loudly visiting with each other in the chapel before sacrament meeting begins and after it ends. I tend to come on the side of being for it because of the spirit of fellowship it can bring into the meeting.
But what about the people sitting in front of, behind and on either side who we don't chat with before the meeting? I perceive that we are in a different stage of life or socio-eceonomic class, I have reason to believe we wouldn't get along very well, or I am naturally shy, etc. What if there was a chance to offer a Sign of Peace to everyone around regardless of my own little cliques?