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Suppose that we only see small “k x k snapshots” of a random two-dimensional “n x n picture”: can we piece the original picture back together? Motivated by the one-dimensional problem of shotgun sequencing DNA, Mossel and Ross raised several interesting questions (like the one aforementioned) about reconstructing��random structures from “small snapshots” in two (and higher) dimensions. In this talk, I will sketch how we can now answer some of these two-dimensional reconstruction questions: in particular, it turns out that the answer to the problem mentioned above exhibits somewhat surprising “two-point concentration,” and getting to this answer involves a combination of entropic methods and tools from percolation.