The first half of the book presents the library Epsilon that has been built up for symbolic polynomial elimination and decomposition with (geometric) applications. It has 8 modules and contains more than 70 functions, which allow one to triangularize systems of multivariate (differential) polynomials, decompose polynomial systems into triangular systems of various kinds (regular, normal, simple, irreducible, or with projection property), decompose algebraic varieties into unmixed or irreducible subvarieties, decompose polynomial ideals into primary components, factorize polynomials over algebraic extension fields, solve systems of polynomial equations and inequations, and handle and prove geometric theorems automatically. The usefulness of the software tools are demonstrated in the second half of the book by a number of selected applications, ranging from qualitative analysis of differential equations in pure mathematics to geometry theorem discovering in automated reasoning and to surface blending and offsetting in computer-aided geometric design. These applications are illuminated with many examples and illustrations. The compact disk distributed with this book contains the entire Epsilon library with documentation, examples, and Maple worksheets.