Recently, the 1701 Act of Settlement has been in the news! See here and here. That act prevents a Catholic (Roman or Eastern Rite) from being the monarch or even being married to the monarch. The Official Website of the British Monarchy explains it thus:
The Act of Settlement of 1701 was designed to secure the Protestant succession to the throne, and to strengthen the guarantees for ensuring parliamentary system of government. The Act also strengthened the Bill of Rights (1689), which had previously established the order of succession for Mary II’s heirs. . . .
According to the 1701 Act, succession to the throne went to Princess Sophia, Electress of Hanover (James I's granddaughter) and her Protestant heirs. However, Sophia died before Queen Anne, therefore the succession passed to her son, George, Elector of Hanover, who in 1714 became King George I. The act was later extended to Scotland as a result of the Treaty of Union enacted in the Acts of Union of 1707.
The Act also laid down the conditions under which alone the Crown could be held. No Roman Catholic, nor anyone married to a Roman Catholic, could hold the English Crown. The Sovereign now had to swear to maintain the Church of England (and after 1707, the Church of Scotland).
In the interests of non-discrimination, some members of Parliament have been suggesting that restriction should be removed. No other religious identity prevents one from being monarch or spouse--should/could a practicing Jew really swear to maintain the Church of England--or an avowed atheist? Recently Nick Clegg, the Deputy Prime Minister had started negotiations to remove this restriction, but the effort has been stopped.
Damian Thompson, linked above, believes that Queen Elizabeth II herself may have stopped it:
"The Queen takes very seriously her coronation oath to uphold the Protestant religion. I wouldn’t dream of accusing her of anti-Catholic prejudice, but suspicion of Rome runs deep in her family. Matters have not been helped by Pope Benedict’s creation of a new ecclesial structure for departing members of the Church of England: the Ordinariate did not play well in royal circles. I suspect that a message has been conveyed to the Lambeth Palace about the inadvisability of changes to the Act in the current climate. And so we are left with a situation in which, if Prince Harry falls in love with a Catholic, he will have to step out of the line of succession – but if the girl is a Scientologist or a practising witch, that’s no problem."
Some have noted that this Act is the one thing that maintains Christianity as the official religion of Great Britain. Some might also say that it hasn't been very effective!