| Gitin
There are certain rabbinic debates where one can only say thank G-d that we do not follow a certain view. Such would be the case in the debate regarding the definition of a mamzer. Halacha has accepted the view that in order to be declared a mamzer one must be the product of an adulterous or incestuous relationship. “Merely” violating a negative command, such as a kohen marrying a divorcee, would not render any offspring mamzerim, though the children would lack the status of kohanim themselves (yes one can have a father who is a kohen and yet not be a kohen oneself). Thankfully Jewish law rejects the view of Rabbi Akiva who declares that any child born of a prohibited union - irrespective of the degree of prohibition - is a mamzer
More significantly all agree (see Kiddushin 68b) that in the case of a child born to a woman who does not observe the laws of family purity, lack of which involves a prohibition of karet, excision, the child of such a union is not a mamzer.  
A similar, actually greater, sigh of relief can be had that the position of Rav Moshe Feinstein outlined below has been accepted as we confront the sexual mores of today.
“He divorced his wife and then spent the night with her at an inn; Beit Shammai says, ‘She does not require a second bill of divorce from him.’ Beit Hillel says, ‘She requires a second bill of divorce from him.’" (Gittin 81a)
As we will soon learn when we begin the study of Masechet Kiddushin next week, there are three ways in which one legally marries[1]. While today the universal practice is to marry with (shaveh) kesef, by the transfer of a ring from the bridegroom to the bride, one can marry by having two people witnessing the man giving the woman a setar, a legal document stating that through this document they are married[2]. This is derived from the fact that the laws of marriage and divorce are written together[3] and just as a get must be in writing a marriage may be effectuated through the written word.
There is a third way to legally, if not morally, marry and that is through the act of sexual intimacy. When one divorces one’s wife yet subsequently spends the night with her at an inn, Beit Hillel argues that the wife would require a second get. This is based on the assumption that once this formerly married couple are seen to enter a hotel room together it is most reasonable to assume that they were intimate. Based on the principle that “ein adam oseh beelato beelat zenut,” a person does not engage in illicit sexual relations when such relations can be non-promiscuous, we declare that the intimate relations between the couple are a form of marriage rather than illicit sexual relations. While Beit Shammai disagree and claim that people have no qualms about engaging in illicit sexual relations the halacha, as it almost always does, follows the view of Beit Hillel.
One of the fiercest and most important halachic debates of the 20thcentury revolved around the application of this principle to the changing sexual ethic. This crucial debate focused on whether a Jewish couple who married civilly - or who lived together absent any formal wedding - required a Jewish divorce to end the marriage. Rav Yosef Eliyahu Henkin [4] argued absolutely. The fact that the couple live together and are intimate with each other means that they are married. There seems to be no better example of the principle that “a person does not engage in illicit sexual relations.”
Yet Rav Moshe Feinstein while agreeing a get would be preferable, ruled that even absent a get the woman is free to remarry. The fact that the couple has chosen a civil marriage - or no maariage at all - indicates that they do not want to be married according to Jewish law. With so many couples living together without any form of marriage it is clear that people are willing to engage in what Jewish law considers illicit sexual relations. Rav Moshe extends his argument even to Jewish marriages which are not conducted according to the strictures of halalcha, allowing re-marriage absent a get, something Rav Henkin vehemently disagreed with.  
While Reform and Conservative Rabbis are often bitter about this rejection of the validity of their marriages[5] it is this ruling which has preserved the unity of the Jewish people. By allowing remarriage without benefit of a get Rav Moshe effectively freed thousands upon thousands of agunot, as many who marry under non-Orthodox auspices fail get a Jewish divorce upon termination of their marriage. It is this ruling which allows Jews from all walks of life to marry each other.
[1] We are referring to the first stage of a Jewish wedding Eirusin, where one was considered legally married though the couple would not yet live together until the Nissuin (how times have changed!) which in Talmudic times took place up to a year later. Nissuin is effectuated by the blessings under the Chupah and the Yichud, seclusion, that immediately follows. 
[2] This should not be confused with the Ketuba which is the first pre-nuptial agreement. 
[3] The Torah does not directly discuss the laws of marriage or even the laws of divorce. Rather the focus of the Torah is the prohibition of re-marrying one's first wife after her remarrying and getting divorced from her second husband too. Husband shopping is not something the Torah allows. Of course in order to get divorced a second time one must first marry and divorce the first time. Thus the Torah begins "If a man takes a wife....and he writes her a bill of divorce.” 
[4] He along with Rav Moshe Feinstein were the two greatest halachic authorities to ever live in America. He is (sadly) much less well known today. 
[5] It is important to note that Rav Moshe ruled that we must try and have a get issued. Only when that is not possible should this ruling be relied upon.