Our ancestors, the Neanderthals

Neanderthals and Cro-Magnons shared Europe for less than 20,000 years. The estimates of total Neanderthal population vary from as many as 70,000 to as few as 3,000.

The Neanderthals used the Mousterian technology and, in western Europe, were improving on it: there were localities in France using Chatelperronian and in Italy using Uluzzian improvements. In contrast, the Cro-Magnons were still using the more primitive Acheulian technology developed by H. Erectus.

50,000 y.a. __Neanderthals met the Cro-Magnons arrriving in the Caucasus.
Neanderthals taught the Cro-Magnons the Mousterian after they arrived in Europe. The Cro-Magnons then used Mousterian for several thousands years.
38,000 y.a. __Cro-Magnons developed the Aurignacian stone tool technology.
After the Cro-Magnons learned the Mousterian, they improved on it developing the Aurignacian technology from 38,000 to 29,000 years ago. The Cro-Magnons spread the Aurignacian all across Europe, including teaching it to the Neanderthals.
It was a continent-wide cultural exchange.
38,000 y.a. __Neanderthals inter-bred with Cro-Magnons
DNA shows there was significant inter-breeding with Neanderthals near the Caucasus about this time. The result is that present-day humans of European stock have
two to four percent Neanderthal DNA.
  A high percentage of the DNA of Neanderthal origin is the Caucasus variety from the area of first contact. It is estimated that 70% of our immune system is inherited from the Neanderthals. Some of the fossils from the Pestera cu Oase cave in Romania are 50% Neanderthal.
  The Cro-Magnon population was very low at the start. It would take only a few intermarriages to achieve a high percentage. As the tribes continued to migrate northwest, continued intermarriage would result in the DNA percentage growing slowly until all the Neanderthals were intermarried and absorbed into the Cro-Magnon tribes. Then, continued arrivals of more Cro-Magnons would dilute the percentage and it would slowly decrease. This means that the early percentage was much higher than the current 2-4%. This leads us to one almost inescapable conclusion:
The Neanderthals did not go extinct.
They were completely absorbed by the Cro-Magnons.

A number of suggestions have been made as to how the technology exchange and the DNA exchange happened. They include rape, kidnapping, etc., but none of these would provide anywhere near enough DNA to get to 4% and they would not achieve the inter-group teaching that clearly occurred.

Neanderthal family

One reasonable explanation is that it may have been a result of exogamous, patrilocal marriage practices generally used by hunter-gatherers world-wide, including both the Cro-Magnons and Neanderthals.

In these inter-marriages, Neanderthal brides would marry Cro-Magnon men. They would come to live with the husband's family and bring her knowledge of the Mousterian with her. She would teach the husband's family the better methods. This was how the Cro-Magnons first learned the Mousterian. Later on, after the Aurignacian was developed, Neanderthal husbands learned the improved Aurignacian technology from their Cro-magnon wives. This hypothesis provides a workable mechanism for significant amounts of both the cultural exchanges and the DNA exchanges to have taken place. It also suggests the reason for the disappearance of the Neanderthals could be that they interbred with the Cro-Magnons until there were no Neanderthals left.