A bastard is defined as a child born from prohibited relationships. Specifically, this term applies to a child born from an adulterous relationship. The primary relationships that result in a bastard are those between a married woman and a man who is not her husband. Close relatives whose union is explicitly forbidden by the Torah, such as a man, his sister, or his mother

In the Babylonian Talmud, it is stated by the sages that even a Jewish bastard who is not allowed to marry a wife (unless she is a slave), if he studies Torah for heaven’s sake, is considered as if he is a high priest who enters the Holy of Holies. A non-Jewish bastard is allowed to marry a non-Jewish woman. And certainly, if he is a Noahide who learns the seven commandments given at Mount Sinai, this act elevates him spiritually far above and beyond the rank of a mere Gentile. Everyone has a free choice whether to be a good person, which means to do G-d’s will by fulfilling His commandments and behaving nicely towards people, or to choose evil, which means doing the opposite of G-d’s commandments and having negative behavior towards others. Overall a bastard is permitted to marry a woman not to mention if he is a Noahide as such he is spiritually elevated, which can have a good effect on his character traits, and certainly can marry a woman.

*This is from a series of articles by Rabbi Bernstein Moshe.