If a group is given by generators and relators, we cannot say much, in general, about its structure. For example, the problem if such a group is non-trivial is algorithmically undecidable. Small cancellation conditions are a well-known class of conditions on group presentation that allows one to study the structure of a given group. I will tell how we use small cancellation groups and their direct limits to prove that Burnside groups of large enough odd and even exponent are infinite. In particular, our approach reduces the exponent (odd case) of the infinite Burnside group to 557. Also, I will give a short survey of our approach to small cancellation theory for rings and show its connections to the classic notion of the Groebner basis. The first part of the talk is based on a joint work with Eliyahu Rips and Katrin Tent, the second part is based on a joint work with Eugene Plotkin, Alexey Kanel-Belov and Eliyahu Rips.