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The subject of this talk is the problem of resolution of singularities in algebraic geometry, but it is intended for a more general audience than just specialists in birational geometry. In the rst part of the talk we will give an introduction to the problem which should be understandable for a general mathematical public. If we cover $X'$ by ane charts, the problem becomes one of parametrizing pieces of $X$ by small pieces of the Euclidean space $k^n$. Algebraically, this localized version of the problem can be interpreted as follows. Let $R$ be $k$-algebra of nite type without zero divisors and let $R$ be a valuation ring containing R and having the same eld of fractions as $R$. Find a smooth nite type k-algebra $R'$ such that $R\subset  R'\subset  R_1$. This existence of such an $R'$ is known as the Local uniformization theorem; it is known when char $k=0$ and is one of the central open problems in the eld when char $k=p>0$. In the second part of the talk we will give a self-contained treatment of valuation theory and its interpretation in birational geometry. We will dene valuations from scratch, give examples (with special attention to valuations centered on algebraic surfaces), dene the space of Zariski-Riemann and explain its connection to the problem of resolution of singularities and local uniformization. In the end of the talk we will discuss a more general notion of universal Puiseux series associated to $R$ (series which need not have well ordered support) as well as their possible applications to the problem of Local uniformization in arbitrary characteristic.