I often hear that the reason we share the Eucharist, the reason we participate in communion, is that Jesus said “do this in remembrance of me.”
While it is true that he did say so, the only two (three) places in which this is found, Luke 22:19 and 1Cor. 11:24.25, the reference is not the sharing, the consummation of communion, but the actions of Christ. To reference liturgical scholar Dom Gregory Dix, these actions, in their ‘bulked up’ four-part version, are (1) taking/offering bread and wine (the Offertory); (2) blessing, giving thanks (the Eucharistic prayer); (3) breaking the bread (the Fraction); and (4) distributing the elements (the Communion). We consume the elements in answer to the remembrance, but this is not the remembrance in itself. The remembrance is to do what Christ did.
The question then becomes: When Christ uttered these words to the Apostles (“do this in remembrance of me”), was he addressing them as Apostles or as Christians? The former is the interpretation commonly favoured by Catholics (including, but not limited to, Roman Catholics and Orthodox), and it ties to the question of when a Eucharist is a Eucharist. Is it a Eucharist when it is ‘performed’ by someone who is not ordained? That is an important question.