  We show how Fermi liquid theory results can be systematically recovered using
a renormalization group (RG) approach. Considering a two-dimensional system
with a circular Fermi surface, we derive RG equations at one-loop order for the
two-particle vertex function $\Gamma $ in the limit of small momentum (${\bf
Q}$) and energy ($\Omega $) transfer and obtain the equation which determines
the collective modes of a Fermi liquid. The density-density response function
is also calculated. The Landau function (or, equivalently, the Landau
parameters $F_l^s$ and $F_l^a$) is determined by the fixed point value of the
$\Omega $-limit of the two-particle vertex function (${\Gamma ^\Omega }^*$). We
show how the results obtained at one-loop order can be extended to all orders
in a loop expansion. Calculating the quasi-particle life-time and
renormalization factor at two-loop order, we reproduce the results obtained
from two-dimensional bosonization or Ward Identities. We discuss the
zero-temperature limit of the RG equations and the difference between the Field
Theory and the Kadanoff-Wilson formulations of the RG. We point out the
importance of $n$-body ($n\geq 3$) interactions in the latter.
