Title: Genus two tropical curves and their Igusa invariants Abstract: An abstract tropical curve is a metric graph where each vertex is equipped with a non-negative integer weight and every zero weight vertex has degree at least three. Their classification by combinatorial type resembles the stratification of the compact moduli space of curves by dual graphs. Tropicalizations of algebraic curves strongly depend on choices of coordinates and the resulting embedded tropical curves become a shadow of the true curves. Providing elementary ways of detecting bad choices of coordinates and "repairing" methods is a current active research area in this field. In this talk, I will discuss such methods for genus 2. Our motivation comes from elliptic curves over an algebraically closed field, which are classified up to isomorphism by the j-invariant. For appropriate choices of coordinates, the tropicalization of an elliptic curve with bad reduction gives a balanced piecewise-linear graph with genus 1, whose distinguished cycle has length equal to negative the valuation of the j-invariant. We generalize this study to curves of genus 2, which are classified by their three Igusa invariants. Contrary to the genus one case, the valuations of these invariants are not sufficient to capture the complete tropical information. I will present an alternative set of invariant which appear to be more natural from the tropical perspective. This is joint work in progress with Hannah Markwig and Ralph Morrison.