One of the themes of Shavuot is the subject of conversion. The entire nation converted on הר סיני as they accepted the Torah. We also read the Book of Ruth which is also about the most famous convert among the Jewish people.
The Rambam discusses the rules for conversion in הלכות איסורי ביאה chapter fourteen. He uses Ruth as the example of one who is מתאמצת, insistent or persistent in her desire to become Jewish. We first try to determine what the motivation was of the potential convert. We first want to make sure that there isn’t a romance involved. It is obvious from ancient times that people will convert in order to marry the Jew they have fallen for in love. We also want to be sure that they are not using the conversion for personal gain.
If after all of this, the convert still stubbornly wants to be Jewish, he is allowed to convert.