The Burnside problem: history and new results

In 1902 W. Burnside asked whether a finitely generated group such that its every element has a finite order is finite. A more restricted question also makes sense: if a finitely generated group with identity $x^n = 1$ is finite. For big enough $n$ (odd $n\ge 557$ and even $n\ge 8000$) the answer is negative, for $n = 2, 3, 4, 6$ the answer is positive, for the remaining exponents the answer is unknown. This problem has a direct connection with hyperbolic groups and negative curvature. I will speak about the latest results, explain the methods, and tell where else one can apply these methods.