Is there any way for the descendants of a mamzer to ever become Jews with full rights, or are they stuck with mamzerus forever? For example, if a man who is a mamzer marries a non-Jew, and then their children convert to Judaism, are the children mamzerim?
1800 years ago, it sounds like the Mishnah would permit a mamzer to marry a non-Jewish (actually, more like quasi-Jewish) slave woman; when that woman is freed she (and her unborn children) become full Jews. A (male) mamzer would thus be allowed to father a non-mamzer child this way.
In contemporary times you can't just go out and buy a quasi-Jewish slave woman (at least not legally in the US; and we don't condone breaking the law). One suggestion has been to find a woman interested in converting to Judaism, but instead of going through full conversion, declare it "conversion to quasi-Jewish slave status"; many rabbis feel this doesn't work as there is no such thing in our location and age.
If a mamzer married a non-Jewish woman, and she and the children then converted, the child(ren) would not be labeled a mamzer, though we generally don't allow a Jewish man to go marry a non-Jewish woman (even if she has plans to later convert). There are rumors that certain rabbis have actually recommended this in certain cases, though. (Again, ask your local rabbi!)
Lastly, Rabbi Moshe Feinstein had the fascinating opinion that mamzer status is transmitted only through normal procreation; a mamzer could choose to marry a normal Jewess and have children only via artificial insemination. I don't think Rabbi Feinstein's opinion on this is commonly accepted. (As heard from a yutorah shiur by Rabbi Willig.)