We provide an overview of recent joint work with Jack Hall, David Rydh and more recently Daniel Halpern-Leistern on the local structure of algebraic stacks. These results rely on the property of coherent completeness which may be viewed as a generalization of Grothendieck's Formal GAGA Theorem categorizing coherent sheaves on a scheme proper over a complete local noetherian ring in terms of compatible families of coherent sheaves on the thickenings of its central fiber. We will briefly mention applications to the existence of moduli spaces in joint work with Daniel Halpern-Leistern and Jochen Heinloth before turning to more recent joint work with Jack Hall and David Benjamin Lim on partial progress extending certain coherent completeness results to positive characteristic.

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