  Li\'enard equations of the form $\ddot{x}+\epsilon f(x)\dot{x}+x=0$, with
$f(x)$ an even function, are considered in the weakly nonlinear regime
($\epsilon\to 0$). A perturbative algorithm for obtaining the number, amplitude
and shape of the limit cycles of these systems is given. The validity of this
algorithm is shown and several examples illustrating its application are given.
In particular, an ${\mathcal O}(\epsilon^8)$ approximation for the amplitude of
the van der Pol limit cycle is explicitly obtained.
