In this talk I will introduce the notion of exponential sums modulo p^m and the conjecture of Igusa about it. I will say about  the development of this conjecture along history due to the works of many mathematicians as Igusa, Denef, Sperber, etc. Moreover, I will present some new ideas and new results  about this conjecture related to my joint works with other mathematicians as Raf Cluckers, Wouter Castryck, etc. In particular, I will show an equivalence between this conjecture on level of log-canonical threshold and Numerical data conjecture associated to any embedded resolution of singularities. I also introduce the new achievement about the Mustata's proof for Numerical data conjecture on level of log-canonical threshold by using a big result of Birkar, Cascini, Hacon and McKernan about the existence of minimal models for varieties of log general type.