Automorphic forms are well-behaved functions on a group (such as GL(n,R)) which are invariant under the action of a discrete arithmetic subgroup (such as SL(n,Z)). They encode vital arithmetic information. Voronoi formulas are Poisson-style summation formulas for automorphic forms. They have been powerful tools in analytic number theory for a long time (with applications to the divisor problem, the circle problem, subconvexity of L-functions, etc). I will talk about Voronoi formulas and their recent development and applications. Some results are joint work with Stephen D Miller, Eren M Kiral.

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