The aim of this paper is to apply polynomial approximation by sums of squares in several real variables to the multidimensional moment problem. The general idea is to approximate any element of the positive cone of the involved function space with sums whose terms are squares of polynomials. First, approximations on a Cartesian product of intervals by polynomials taking nonnegative values on the entire R2, or on R+2, are considered. Such results are discussed in Lμ1R2 and in CS1×S2-type spaces, for a large class of measures, μ, for compact subsets Si, i=1,2 of the interval [0,+∞). Thus, on such subsets, any nonnegative function is a limit of sums of squares. Secondly, applications to the bidimensional moment problem are derived in terms of quadratic expressions. As is well known, in multidimensional cases, such results are difficult to prove. Directions for future work are also outlined.