Ok this topic is unfathomably huge. The Mongol empire lasted over 100 years and encompassed most of Eurasia. To further complicate things it split into four different sucessor states each with its won unique character.Thus you could literally write about a dozen books on this. Also by culture do you mean the arts? Or culture more generally as in societal ethos? So to make my life easier I'm going to assume you mean the arts (painting, poetry, etc) and focus on my area of expertise the Ilkhanate, a Mongol sucessor state that encompassed Iran plus areas of Iraq, Afghanistan, Pakistan, Azerbijan and Uzbekistan. Later this evening /u/Jasfss will hopefully turn up to talk about China in far more depth than I can. Though if he doesn't I'll post what I know.

Their biggest impact was on the art world. However this was not so much in terms of them introducing Mongol motifs into art, instead they brought in motifs from other areas they conquered and came into contact with. So in Persian paintings we see landscapes painted in Chinese styles. Another good example is the Universal History/Compendium of chronicles which was compiled by Rashid al-Din, a Persian serving as a government minister for the Mongols. This manuscript was rather beautifully illustrated, and we can see a wide variety of styles influencing the art such as Buddhist, Siennese Italian, Chinese and Byznantine iconography. If you look at this and this, both from 1314 copies of the work, you can see that the illustrations have some rather Chinese influences. These manuscripts would would have been illuminated in government workshops in Tabriz which had artists from across the Empire working in them. It's also interesting to note this influence was not one way. There's a good article called Siena on the Silk Roads by Roxxan Prazniak which examines the influence of Ilkhanate art on Siennese art.

If you want more on things like scholarly study and exchange of knowledge in the Mongol Empire feel free to ask further questions.