This volume is about the most mind-boggling sacrament of the Christian faith, also referred to as the Sacrament of the Altar, the Eucharist: in its Roman Catholic interpretation, the conversion of the substance of the bread and wine into the body and blood of Christ for Holy Communion. The challenge of providing a rational interpretation of this doctrine of faith proved to be one of the most contentious issues in the Western history of ideas, apparently going against self-evident metaphysical principles (requiring accidents existing without a substance, and a body in several places at the same time, etc.), and dividing schools of thought, indeed, eventually, warring religious factions. The volume addresses both the metaphysical, theoretical issues involved in this challenge and the historical, theological developments of how meeting this challenge played out first in the schools and even later in religious schisms, leading to the paradigmatic shift from medieval to modern forms of thought. The essays of the volume derive from the lectures of an eponymous international conference held in Budapest, Hungary, which was also the occasion of founding the Society for the History of European Ideas (SEHI); accordingly, the book is the first volume of the annual Proceedings of the SEHI. This book is aimed just as much at laymen and religious scholars seeking a better understanding of their faith as at anyone seeking this understanding with a non-religious attitude.