The task of solving non-linear polynomial equations has many applications in science and technology. Macaulay, Buchberger and others developed the theoretical ground to solve them in the 1930's and 60's via Groebner bases and elimination theory. More recently, Lazard, Faugere and others reframed the problem in linear algebra terms, allowing important improvements in the algorithmic complexity. In this course we walk this path. First, we develop the necessary theory from algebraic geometry. We then focus on the relation between Groebner bases and linear algebra. And finally, we discuss the complexity of the algorithms to compute Groebner bases using linear algebra.