  We describe the moduli stack of Gushel-Mukai varieties as a global quotient
stack and its coarse moduli space as the corresponding GIT quotient. The
construction is based on a comprehensive study of the relation between this
stack and the stack of Lagrangian data; roughly speaking, we show that the
former is a generalized root stack of the latter. As an application, we define
the period map for Gushel-Mukai varieties and construct some complete
nonisotrivial families of smooth Gushel-Mukai varieties. In an appendix, we
describe a generalization of the root stack construction used in our approach
to the moduli space.
