A recent article from the BBC mentions two studies looking at the Neanderthal genome and it’s influence over present day non-African genes. This strongly indicates that interbreeding did exist between the different species and some fertile offspring had been produced, although the descendents became less fertile over time.
By studying the genome neanderthal versions of genes were present including a gene variant associated with the difficulty to stop smoking. Other genes appeared to give humans an advantage to the cooler conditions including proteins to toughen the skin and hair. However there was also evidence to show that the Neanderthals passed on gene markers that increased and decreased the risk of Crohn’s disease.
More a more detailed review can be found in Science Magazine which is definitely worth a read. See below for the full citations for the genome studies.
A genome sequence of a Neanderthal showing interbreeding:
Prufer, K., F. Racimo, et al. (2014). “The complete genome sequence of a Neanderthal from the Altai Mountains.” Nature 505(7481): 43-49.
Neanderthal genes in modern humans, including contribution to skin pigmentation:
Vernot, B. and J. M. Akey (2014). “Resurrecting Surviving Neandertal Lineages from Modern Human Genomes.” Science.