Polynomials over finite fields play a central role in algorithms for cryptography, error correcting codes, and computer algebra. The complexity of multiplying such polynomials is still a major open problem. Let p be a prime, and let M-p(n) denote the bit complexity of multiplying two polynomials in F-p[X] of degree less than n. For n large compared to p, we establish the bound Mp( n) = O(nlogn8(log*n) log p), where log* n = min{k. epsilon N : log...kx log n <= 1} stands for the iterated logarithm. This improves on the previously best known bound M-p( n) = O( nlogn loglogn log p), which essentially goes back to the 1970s.