This chapter presents algebraic synthesis and algebrogeometric approaches in the study of control problems rely on the theory of invariants and canonical forms of polynomial matrices and use the theory of Groebner bases for the computation of solutions of polynomial equations in many variables. This chapter provides first a classification of the different types of algebraic computations, according to whether they can be performed symbolically and/or numerically, and then considers three representative families of algebraic control theory problems, which require the use of symbolic computation tools.