On the two-dimensional square lattice, assign i.i.d. nonnegative weights to the edges with common distribution $F$. For which distributions $F$ is there an infinite self-avoiding path with finite total weight? This question arises in first-passage percolation, the study of the random metric space $\mathbb{Z}^2$ with the induced random graph metric coming from the above edge-weights. It has long been known that there is no such infinite path when $F(0) < 1/2$ (there are only finite paths of zero-weight edges), and there is one when $F(0) > 1/2$ (there is an infinite path of zero-weight edges). The critical case, $F(0) = 1/2$, is considerably more difficult due to the presence of finite paths of zero-weight edges on all scales. I will discuss work with W.-K. Lam and X. Wang in which we give necessary and sufficient conditions on $F$ for the existence of an infinite finite-weight path. The methods involve comparing the model to another one, invasion percolation, and showing that geodesics in first-passage percolation have the same first order travel time as optimal paths in an embedded invasion cluster.