Return of the Romanovs?
Rumors flew... without bodies to show, many refused to believe that the Romanovs were dead. When a memorial service was held for Czar Nicholas in London, George V refused to attend or send a representative; many interpreted this as a sign that George V knew the Romanovs were still alive.
Some said that the Bolsheviks had lied about the execution, and that they had secretly released the Romanovs from Russia presumably to honor an unspecified peace treaty with Germany and to gain "sympathy for their cause in the West" [quoted from Strange Stories, Amazing Facts]; of course, they could hardly gain sympathy for the Czar's release if they also continued to claim that he had been execuited, so this second claim was unreasonable. Others said the royal family had escaped to Poland. It was asserted that only Dr. Botkin and the servants had been killed in the basement of Ipatiev House, both to simulate the death of the Romanovs and to silence the witnesses to the family's exile.
In 1919 a book entitled 'Rescuing the Czar' was published, in which the author, a self-described 'American secret agent' named James P. Smyth, explained how he guided the Romanovs through a secret tunnel to the British Consulate in Ekaterinburg, from where they were "spirited" out of Russia to Tibet. The book had little to substantiate it's claims, and serious researchers and historians never treated it as a true account.
As time passed, the focus of the rumors and claims changed; fewer and fewer people believed that the whole Romanov family had escaped... but more and more believed that maybe just one or two of the children could have. These beliefs, combined with rumors of a fabulous Romanov fortune supposed to be held in Western banks, led some people to claim they were surviving children of the Russian royal family.
The first serious claim was made in 1922 in Berlin by a woman being held in an insane asylum. She stated that she was Grand Duchess Anatasia, youngest daughter of the Romanov royal family, and claimed to have been smuggled away from the execution to the Balkens by a Bolshevik soldier; later, after the soldier died, she came to Berlin where she was placed in the mental hospital after attempting suicide. Controversy raged around her claim for decades, as no definitive proof for or against it came forward... until 1994, years after her death, when DNA evidence finally proved once and for all that she could not have been the Grand Duchess Anastasia. [See also: Mysteries Article: Anna Anderson... a.k.a. Anastasia?]
The second notable claim of Romanov descent came in 1961, when Colonel Michael Goleniewski, a Polish officer that had recently defected to the United States, claimed that he was the only son of the Romanov royal family, Grand Duke Alexis. His claim is said to have been supported by "a former research director of the CIA" (according to Mysteries of the Unexplained); and -- according to Strange Stories, Amazing Facts -- two women signed the guest book at his wedding in 1964 as Olga and Tatiana, the names of the former Czar's two oldest daughters. Goleniewski claimed to have exposed some 200 Russian agents to the United States government because of his hatred for Communism... but, in the end, his claims were never taken seriously for one simple reason. Alexis Romanov had suffered from hemophilia, an uncurable genetic disorder; Goleniewski didn't.
Over the years, Ipatiev House in Ekaterinburg had become a place of pilgrimage for Russian "Monarchists"... unamused, the Russian government ordered it destroyed in 1977. But despite this attempt to bury an unpleasant reminder of Communist Russia's rise to control, the site was marked again by those who would not forget the execuition with a simple Russian catholic cross.

End of the Mystery?
Though the rest of the world didn't know it, the Bolsheviks had indeed lied about the execution of the Romanovs... but not in the way expected. The royal family had in fact been shot, but then they were dumped into the mine, not the swamp. Forty-eight hours later, nervous that the corpses would be too easily found, Yurovski had the bodies retrieved, and then tried burning them. Only two bodies were burned... the two smallest, Alexis' and Anastasia's; both bodies were said to be reduced to ashes, though some believe the burning story might have been a coverup for an escape by the two children. In any case, it was decided that burning all the bodies was too much trouble, and, instead, the remaining bodies were then buried in a mass grave in the middle of a dirt road. Ironically, the White Army had almost certainly walked across the burial site several times while searching the swamps and Four Brothers Mine for the dead royalty.
In 1979, the final grave of the Romanovs was finally found by a local historian and a Soviet TV personality; but it was not a good time, politically speaking, to announce the finding of the missing royal family. So two skulls were excavated, analyzed, and re-buried, and the discovery was kept quiet. Ten years later in 1989, under the aegis of Glasnost, the discovery was finally announced to the world press.
In July 1991, the skeletons of the Romanovs and their attendants were finally, officially, excavated and recovered. After the bones had been sorted and examined, all agreed... there were only nine complete skeletons. The two missing skeletons belonged to two of the Romanov children; the two youngest, Alexis and Anastasia. In October, 1994, the results of DNA testing used to compare the DNA of the skeletons to the DNA of known living relatives of the Romanovs was announced, and, once and for all, the skeletons were confirmed as being those of the missing royal family and their attendents.
But what of Alexis and Anastasia? Where there are no bodies, there is room for speculation; and while it is unlikely young Alexis would have survived long with hemophilia (which prevents blood from clotting, and so produces unstoppable bleeding), some do still believe that young Anastasia could have escaped the massacre that claimed the rest of the last royal family of Russia.