  Completing a strategy of Gou\"ezel and Lalley, we prove a local limit theorem
for the random walk generated by any symmetric finitely supported probability
measure on a non-elementary Gromov-hyperbolic group: denoting by $R$ the
inverse of the spectral radius of the random walk, the probability to return to
the identity at time $n$ behaves like $C R^{-n}n^{-3/2}$. An important step in
the proof is to extend Ancona's results on the Martin boundary up to the
spectral radius: we show that the Martin boundary for $R$-harmonic functions
coincides with the geometric boundary of the group. In an appendix, we explain
how the symmetry assumption of the measure can be dispensed with for surface
groups.
