Green's function$G(x)$ of a zero mean random walk on the $N$-dimensional integer lattice ($N\geq2$) is expanded in powers of $1/|x|$ under suitable moment conditions. Inparticular, we find minimal moment conditions for$G(x)$ to behave like aconstant times the Newtonian potential (or logarithmic potential in twodimensions) for large values of $|x|$. Asymptotic estimates of $G(x)$ in dimensions $N\geq 4$, which are valid even when these moment conditions areviolated, are computed.Such estimates are applied to determine the Martinboundary of the random walk. If $N= 3$ or $4$ and the random walk has zero meanand finite second moment,the Martin boundary consists of one point, whereas if$N\geq 5$, this is not the case, because non-harmonic functions arise as Martinboundary points for a large class of suchrandom walks. A criterion for whenthis happens is provided.

1991 Mathematics Subject Classification: 60J15,60J45, 31C20.