About seven years ago my wife and I (who are childless) left our old church and set out to find the best church we could in the area. We ended up in a church that practices paedo (child) communion. They do not practice this because they believe that the children are saved, but because they believe that the children are members of God's Covenant People because they have been baptized, as most presbyterians do. They often point to the passover feast as the similar Old Testament 'sacrament' where the children ate with the whole family. What other evidence both biblical and historical exists for this practice.
Here's what my denomination, the United Methodist Church, says about child communion:
I understand that other denominations have children wait until they have a better understanding of communion. I'm sure they have good, scriptural reasons for their belief.