What is the Out-of-Africa theory? According to the Out-of-Africa theory, the evolution of humans has happened exclusively in Africa. For the correctness of the theory, two assumtions are required: First, human evolution in Africa must be finished and second all humanoids outside Africa (the Neanderthals, several subraces of Homo Erectus) must have been completely exterminated without a single case of interbreeding.
Inconvenient finds in the Caucasus Until now it was assumed that not just humans, but all hominids have evolved in Africa and have moved to other parts of the world after that. Finds in Georgia (south of the Caucasus-mountains next to Russia) suggest that Homo Erectus first arose in Eurasia and not in Africa. In other words the direction of spreading of Homo Erectus was exactly reversed than previously thought.
If Homo Erectus has not evolved in Africa, that is no proof that Homo Sapiens did not come from Africa, yet the central assumption of the Out-of-Africa theory is shaken because it was thought that all hominids have evolved in Africa and gone to the rest of the world in waves where evolution has stopped. Obviously the true story of the evolution of mankind is a little more complicated than that.
The genetic contribution of Neanderthals outside Africa Modern methods for sequencing of DNA are available only for a few years to find out the truth. Several inquiries have already found Neanderthal-DNA in non-Africans and therefore proven that Neanderthals were not exterminated completely and have contributed to the genetic makeup of non-Africans.
Researchers currently estimate that about 1 to 4% of Eurasian (i.e. non-African) DNA is coming from Neanderthals. However such percentage estimations are very vague. For example the DNA between human and chimpansee is 96% to 99% identical, depending on calculation method. Neanderthals and early forms of Homo Sapiens surely were even more similar than that, that means just a few percent of the DNA were different - after the mixture of these hominids we see many genes which were identical before and which can be not clearly assigned.
Researchers therefore look for so-called "junk-genes", genes without function, to estimate ancestry. These genes without function are passed "neutrally", in other words, they have no influence on the person who has them.
The limits of this method can be made clear with a simple thought experiment: Suppose, there is a group of people in which each is the offspring of a Neanderthal and an early form of Homo Sapiens. These people would have exactly half of the genes from Homo Sapiens and the other half from Neanderthals. By using "junk genes", one would be able to confirm a ratio of 50% pretty accurately.
If however this group of people would settle different parts of the world, the ratio of "junk genes" would hardly change at all, while the "real genes", the genes which contain the blueprint of the human body, would continue to evolve: In tropical areas with Malaria, for example genes for dark skin and resistance against Malaria would become universal, while in northern areas with dairy farming, genes for lactose toleration and light skin would become universal.
After a couple of generations the populations would look very different from each other, would be immune or sensitive to different diseases and would differ genetically in many ways. But a inquiry into "junk genes" would still measure of roughly 50% Neanderthal and 50% early Homo Sapiens for both populations.
"Junk genes" are therefore very useful to proof ancestry, but less useful to answer the question how "similar" modern Eurasians are to Neanderthals.
Summary Because of many inconvinient finds and facts the Out-of-Africa theory was already shaken for quite some time, however there was no final proof of it being wrong. New results, especially DNA-tests from the last years, have now finally refuted the Out-of-Africa theory.