In order to have well defined rules for the perturbative calculation of quantities of interest in an interacting quantum field theory in curved spacetime, it is necessary to construct Wick polynomials and their time ordered products for the noninteracting theory. A construction of these quantities has recently been given by Brunetti, Fredenhagen, and Kohler, and by Brunetti and Fredenhagen, but they did not impose any ``locality'' or ``covariance'' condition in their constructions. As a consequence, their construction of time ordered products contained ambiguities involving arbitrary functions of spacetime point rather than arbitrary parameters. In this paper, we construct an ``extended Wick polynomial algebra''-large enough to contain the Wick polynomials and their time ordered products. We then define the notion of a {\it local, covariant quantum field}, and seek a definition of {\it local} Wick polynomials and their time ordered products as local, covariant quantum fields. We impose scaling requirements on our local Wick polynomials and their time ordered products as well as certain additional requirements-such as commutation relations with the free field and appropriate continuity properties under variations of the spacetime metric. For a given polynomial order in powers of the field, we prove that these conditions uniquely determine the local Wick polynomials and their time ordered products up to a finite number of parameters. (These parameters correspond to the usual renormalization ambiguities occurring in Minkowski spacetime together with additional parameters corresponding to the coupling of the field to curvature.) We also prove existence of local Wick polynomials. However, the issue of existence of local time ordered products is deferred to a future investigation