  Strong laws of large numbers are established for random fields with weak or
strong dependence. These limit theorems are applicable to random fields with
heavy-tailed distributions including fractional stable random fields.
  The conditions for SLLN are described in terms of the $p$-th moments of the
partial sums of the random fields, which are convenient to verify. The main
technical tool in this paper is a maximal inequality for the moments of partial
sums of random fields that extends the technique of Levental, Chobanyan and
Salehi \cite{chobanyan-l-s} for a sequence of random variables indexed by a
one-parameter.
