Valiant has proposed a new theory of algorithmic computation based on perfect matchings and Pfaffians. We study the properties of matchgates--the basic building blocks in this new theory. We give a set of algebraic identities which completely characterizes these objects for arbitrary numbers of inputs and outputs. These identities are derived from Grassmann-Pl?ucker identities. The 4 by 4 matchgate character matrices are of particular interest. These were used in Valiant?s classical simulation of a fragment of quantum computations. For these 4 by 4 matchgates, we use Jacobi?s theorem on compound matrices to prove that the invertible matchgate matrices form a multiplicative group. Our results can also be expressed in the theory of Holographic Algorithms in terms of realizable standard signatures. These results are useful in establishing limitations on the ultimate capabilities of Valiant?s theory of matchgate computations and Holographic Algorithms.