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Abstract: The moduli space of curves and its compactification by stable curves has been a central object of study in algebraic geometry since its inception. Over the past few decades, building on major advances in the minimal model program, an analogous theory of compact moduli spaces of higher dimensional varieties has been developed. These moduli spaces parametrize stable log varieties (X,D), i.e. pairs of a variety X and a divisor D with reasonable singularities (generalizing the notion of nodal curves) and a stability condition. The geometry of these moduli spaces is much more mysterious in higher dimensions and little is known beyond some examples. After giving an introduction to the theory of stable log varieties, I will present joint work with K. Ascher, G. Inchiostro, and Z. Patakfalvi on wall-crossing morphisms for these moduli spaces which describe how the geometry of the compactification changes as one varies the coefficients. I will illustrate these wall-crossings and how they can be used to study the structure of these moduli spaces in the explicit example of rational elliptic surfaces (joint work with K. Ascher).