We consider the d-dimensional cube packing problem (d-CPP): given a list L of d-dimensional cubes and (an unlimited quantity of) d-dimensional unit-capacity cubes, called bins, find a packing of L into the minimum number of bins. We present two approximation algorithms for d-CPP, for fixed d. The first algorithm has an asymptotic performance bound that can be made arbitrarily close to 2 - (1/2)(d). The second algorithm is an improvement of the first and has an asymptotic performance bound that can be made arbitrarily close to 2 - (2/3)(d). To our knowledge, these results improve the bounds known so far for d = 2 and d = 3, and are the first results with bounds that are not exponential in the dimension.403173187