The problem of local uniformization can be seen as the local version of resolution of singularities. For instance, for an algebraic variety $X$ and a point $x\in X$, a valuation $v$ centered at $\mathcal{O}_{X,x}$ admits local uniformization if there exists a proper birational map $X'\rightarrow X$ such that $\mathcal{O}_{X',x'}$ is regular, where $x'$ is the center of $v$ in $X'$. This problem was introduced by Zariski in the 1940's in order to prove resolution of singularities for algebraic varieties. He succeeded in proving local uniformization for valuations on algebraic varieties over fields of characteristic zero. He also proved, using his approach via local uniformization, resolution of singularities in low dimensions. In 1964, Hironaka proved resolution of singularities for any algebraic variety over fields of characteristic zero. However, both resolution of singularities and local uniformization are widely open problems in positive characteristic. In this talk, I will present my joint work with Mark Spivakovsky on the reduction of local uniformization to the rank one case. It was believed for a long time, that in order to prove local uniformization it was enough to prove it for rank one valuations. We proved that this assertion is true for a broad category of noetherian local domains. I will discuss this result as well as present our recent developments for the case of rings which are not necessarily domains.