Abstract: The local uniformization problem is the local version of resolution of singularities for algebraic varieties. In this talk, we will discuss some strategies to solve this problem, which is open in positive characteristic. We will introduce the notion of key polynomials and describe how we hope to use them to improve the known results about local uniformization. Also, we will discuss the differences between the cases of positive and zero characteristic. At the end, we will present a simple example that illustrates these differences.