  These are the notes of five lectures given at the Summer School {\em
Geometric and Topological Methods for Quantum Field Theory}, held in Villa de
Leyva (Colombia), July 2--20, 2007. The lectures are meant for graduate or
almost graduate students in physics or mathematics. They include references,
many examples and some exercices. The content is the following.
  The first lecture is a short introduction to algebraic and proalgebraic
groups, based on some examples of groups of matrices and groups of formal
series, and their Hopf algebras of coordinate functions.
  The second lecture presents a very condensed review of classical and quantum
field theory, from the Lagrangian formalism to the Euler-Lagrange equation and
the Dyson-Schwinger equation for Green's functions. It poses the main problem
of solving some non-linear differential equations for interacting fields.
  In the third lecture we explain the perturbative solution of the previous
equations, expanded on Feynman graphs, in the simplest case of the scalar
$\phi^3$ theory.
  The forth lecture introduces the problem of divergent integrals appearing in
quantum field theory, the renormalization procedure for the graphs, and how the
renormalization affects the Lagrangian and the Green's functions given as
perturbative series.
  The last lecture presents the Connes-Kreimer Hopf algebra of renormalization
for the scalar theory and its associated proalgebraic group of formal series.
