Regions of the genome that have been the target of positive selection specifically along the human lineage are of special importance in human biology. We used high throughput sequencing combined with methods to enrich human genomic samples for particular targets to obtain the sequence of 22 chromosomal samples at high depth in 40 kb neighborhoods of 49 previously identified 100-400 bp elements that show evidence for human accelerated evolution. In addition to selection, the pattern of nucleotide substitutions in several of these elements suggested an historical bias favoring the conversion of weak (A or T) alleles into strong (G or C) alleles. Here we found strong evidence in the derived allele frequency spectra of many of these 40 kb regions for ongoing weak-to-strong fixation bias. Comparison of the nucleotide composition at polymorphic loci to the composition at sites of fixed substitutions additionally reveals the signature of historical weak-to-strong fixation bias in a subset of these regions. Most of the regions with evidence for historical bias do not also have signatures of ongoing bias, suggesting that the evolutionary forces generating weak-to-strong bias are not constant over time. To investigate the role of selection in shaping these regions, we analyzed the spatial pattern of polymorphism in our samples. We found no significant evidence for selective sweeps, possibly because the signal of such sweeps has decayed beyond the power of our tests to detect them. Together, these results do not rule out functional roles for the observed changes in these regions-indeed there is good evidence that the first two are functional elements in humans-but they suggest that a fixation process (such as biased gene conversion) that is biased at the nucleotide level, but is otherwise selectively neutral, could be an important evolutionary force at play in them, both historically and at present.