Holy Thursday - Eucharist, The Body of Christ

Holy Thursday, the night of the Last Supper aka the Lord's Supper, the institution of the Holy Eucharist -- why did Jesus do what he did and say what he said? What did he mean when he said "This is my body and this is my blood" and "Do this in Memory of Me"?  And what does the Catholic Church mean by talking about transubstantiation and the body of Christ?  All this is remembered & celebrated on the Feast of Corpus Christi.