  Parameterized algebraic curves and surfaces are widely used in geometric
modeling and their manipulation is an important task in the processing of
geometric models. In particular, the determination of the intersection loci
between points, pieces of parameterized algebraic curves and pieces of
algebraic surfaces is a key problem in this context. In this paper, we survey
recent methods based on syzygies and blowup algebras for computing the image
and the finite fibers of a curve or surface parameterization, more generally of
a rational map. Conceptually, the main idea is to use elimination matrices,
mainly built from syzygies, as representations of rational maps and to extract
geometric informations from them. The construction and main properties of these
matrices are first reviewed and then illustrated through several settings, each
of them highlighting a particular feature of this approach that combines tools
from commutative algebra, algebraic geometric and elimination theory.
