  In this paper, we investigate the law of large numbers for strictly
stationary random fields, that is, we provide sufficient conditions on the
moments and the dependence of the random field in order to guarantee the almost
sure convergence to $0$ and the convergence in $\mathbb L^p$ of partials sums
over squares or rectangles of $\mathbb Z^d$. Approximation by multi-indexed
martingales as well as by $m$-dependent random fields are investigated.
Applications to functions of $d$-independent Bernoulli shifts and to
functionals of i.i.d.\ random fields are also provided.
