This monograph is concerned with quantitative versions of the notion of rectifiability. Recall that a $d$-dimensional subset of $\mathbf {R}^n$ is called rectifiable if it is contained in the union of countable family of $d$-dimensional $C^1$ submanifolds, except possibly for a set of Hausdorff measure zero. This is clearly a qualitative condition, i.e., there are no bounds involved. Our main result provides the equivalencebetween several conditions which can be viewed as providing a natural definition for quantitative rectifiability. As amusing feature of our methods is the role played by singular integral operators, which provide a bridge for passing between various geometrical conditions. We also obtain a higher-dimensinal version of Peter Jones' travelling salesman theorem.