In this talk, we will discuss the question of whether continuous, simple curves in Euclidean space with sigma-finite length have tangents at any points.  The results on $\sigma$-finite curves that we will discuss were initiated by the observation that the graph of a continuous function on [0,1] that satisfies a weak-Lipschitz property has sigma-finite one-dimensional Hausdorff measure. We will discuss our conclusion that every $\sigma$-finite curve has a tangent, in the pointwise sense, on a set of positive measure. This is joint work with M. Csornyei.