Resultants are computational tools for determining whether or not a system of polynomials has a common root without actually solving for the roots of these equations. Resultants can also be used to solve for the common roots of polynomial systems. Classical resultants are typically represented as determinants whose entries are polynomials in the coefficients of the original polynomials in the system. The work in this dissertation on classical resultants focuses on bivariate polynomials. It is shown that bivariate resultants can be represented as determinants in a variety of innovative ways and that these various formulations are interrelated. Remarkable internal structures in these resultant matrices are exposed. Based on these structures, efficient computational algorithms for calculating the entries of these resultant matrices are developed. Sparse resultants are used for solving systems of sparse polynomials, where classical resultants vanish identically and hence fail to give any useful information about the common roots of the sparse polynomials. Nevertheless, sparse polynomial systems frequently appear in surface design. Sparse resultants are usually represented as GCDs of a collection of determinants. These GCDs are extremely awkward for symbolic computation. Here a new way is presented to construct sparse resultants as single determinants for a large collection of sparse systems of bivariate polynomials. An important application of both classical and sparse resultants in geometric modeling is implicitization. Implicitization is the process of converting surfaces from parametric form into algebraic form. Classical resultant methods fail when a rational surface has base points. The method of moving quadrics, first introduced by Professor Tom Sederberg at Brigham Young University, is known empirically to successfully implicitize rational surfaces with base points. But till now nobody has ever been able to give a rigorous proof of the validity of this technique. The first proof of the validity of this method when the surfaces have no base points is provided in this dissertation.