As has been pointed out in Tony Carbery's answer, a positive polynomial over $\mathbb{R}$ in two or more variables need not be a sum of squares. However, Bill Helton showed in 2002 that this does hold for polynomials in noncommuting variables!
Helton, J. William. "Positive'' noncommutative polynomials are sums of squares. Ann. of Math. (2) 156 (2002), no. 2, 675–694. http://www.jstor.org/stable/3597203.
