I moved this question to its own board because it tends to generate a lot of discussion.
There are 2 basic schools of thought. The one held in academic and scientific circles and by most who study the subject is that no one survived, based on the overwhelming preponderance of evidence of the murders. We have eyewitness testimony that everyone was killed. Subsequent investigations concluded the same. The sheer logic of the situation is clear that even "if" anyone survived the initial shootings, beatings and stabbings, they would have soon died from lack of proper medical care, exposure, and lack of facilities to care for them...Notwithstanding there is the virtual impossibility at the time of hiding an Imperial Grand Duchess needing medical care, food and shelter, for any length of time without discovery.
We know for certain that Tsarevitch Alexei was one of the missing bodies. Given his hemophilia, everyone concedes that he could not have survived long. A fall at Bieloviezhe and later events at Spala nearly killed him...What would bullets and beatings and stabbings do? Besides there is the fact that as heir to the throne, the Bolsheviks would have certainly made quite certain he was dead. There is also an historical account and testimony from a man who saw Alexei's body and described it in great detail.
The other thought says that since 2 bodies are missing the book cannot be "closed" on the possibility of survival. They tend to rely on complicated conspiracy theories and fantasy stretches of reality for support, but find the "possibility" just too compelling to let it go.
We have talked to Dr. Terry Melton who performed some of the original mtDNA analysis comparing Anna Anderson's DNA with the DNA from Alexandra's direct family line and are convinced beyond doubt that her work was accurate, reliable and conclusive that Anna Anderson was in no way related to Empress Alexandra Feodrovna.
The Russian forensic scientists who have examined the remains are themselves clearly convinced that they have the remains of Anastasia and the missing body is in fact Marie Nicholaievna.
There are those, who will probably reply here, with their own preferred version of history.