How was Europe affected by Mongolian contact?

I know most cultures were greatly affected by the influence of the Mongols and Genghis Khan, but it wasn't until after his death (when Ogodei was the Khan) that they moved into Europe. I know there was initial conflict between them, the Turks, and the Georgians. But, that's all I know. I was just curious as to if the Mongols had that big of an impact on Europe.