Hylomorphists coherently commit to intrinsically unified composites only if they affirm the existence of prime matter. Doing so frees them to concede that for a substance y composed of the xs, the causal powers of the xs upwardly determine y's causal powers. Unlike some hylomorphic theories that endorse upward determination, however a theory that includes prime matter is not a version of non-reductive physicalism. Chapter 1 introduces Robert Koons's distinction between Staunch Hylomorphism (SH) and Faint-hearted Hylomorphism (FH). SH is committed to composite substances, but FH holds denies there are such. I argue that Koons's answer to the question of how a substance y's powers are related to those of the xs collapses into a version of FH on account of his view that y's fundamental parts are the "enduring substrate of substantial change."Chapter 2 studies three hylomorphic theories that claim emergent causal powers distinguish composite substances from accidental unities. Both Patrick Toner and William Jaworski insist that emergent powers are not realized in the powers of the xs as the xs are configured in y. In contrast, Eleonore Stump claims that some of y's powers are emergent in a weaker sense, such that they are realized in the powers of the xs. I argue that all three views are instances of FH. Each regards substantial forms as higher-level properties of a sort that do not forge intrinsic unity among a composite's parts. Chapter 3 argues that SH must characterize relations among a substance's properties differently than Toner, Jaworski, or Stump. Unlike the proposals of Toner and Jaworski, SH must be compatible with the view that the properties of the xs upwardly determine all y's properties. Unlike Stump's theory, however, SH must not claim that y's substantial form F is among the upwardly determined properties of y. On the contrary, the properties y bears as a member of its substantial kind are among F's effects. Chapter 4 argues that SH can fulfill these desiderata by appealing to prime matter. If y's substantial form Factualizes a principle more basic than physical matter, i.e., some prime matter M, then F is not among y's physical properties; rather, F's is metaphysically prior to any such property. Moreover, F is the form not only of y, but also y's physical parts, the xs. In that case, being part of y is of the essence of every x, and y's intrinsic unity is secured. Now if F is the form of every x, then for every x, the properties x bears as a functional part of y are also posterior to F. A fortiori, the powers of y that are upwardly determined by the properties of xs are posterior to F. Therefore, SH is compatible with the upward determination of y's causal powers.