Her fame peaked in 1963-64. While making "Cleopatra," she and Richard Burton had a very, very famous affair. I watched my mother ask her mother about it, everyone had an opinion. Later both divorced their spouses (Eddie Fisher had 11 Top-5 songs in the 50s) so they could get married, which took the luster off the affair.
Just as the traditional story is great because it unified Western Civilization's old and new empires in sexual love (Egypt's Pharaoh Cleopatra with Rome's Caesar Marc Antony, symbolizing the transfer of power from the Middle East to Europe), the movie unified the American audience (Taylor) and the British audience (Lionel Barrymore died, Laurence Olivier was getting old, so Burton was now the top young actor there). The highly-publicized affair was "tailor"-made to sell tickets in both countries.
I figured out why critics were disappointed when I was older. In the then-most expensive movie of all time, her version of the Pharaoh wasn't all horny for the Caesar as in the much more enjoyable 1934 Cleopatra movie. Watch them on TV and you'll prefer Claudette Colbert to the snitty, bourgeois American housewife Elizabeth Taylor portrayal. The story had traditionally intersected romanticism with world affairs, but the movie is a creature of its 1964 post-McCarthy conservative time.