  These lecture notes have been written for a short introductory course on
universality and renormalization group techniques given at the VIII Modave
School in Mathematical Physics by the author, intended for PhD students and
researchers new to these topics.
  First the basic ideas of dynamical systems (fixed points, stability, etc.)
are recalled, and an example of universality is discussed in this context: this
is Feigenbaum's universality of the period doubling cascade for iterated maps
on the interval. It is shown how renormalization ideas can be applied to
explain universality and compute Feigenbaum's constants.
  Then, universality is presented in the scenario of quantum field theories,
and studied by means of functional renormalization group equations, which allow
for a close comparison with the case of dynamical systems. In particular,
Wetterich equation for a scalar field is derived and discussed, and then
applied to the computation of the Wilson-Fisher fixed point and critical
exponent for the Ising universality class.
  References to more advanced topics and applications are provided.
