  We determine the Artin-Mazur \'etale homotopy types of moduli stacks of
polarised abelian schemes using transcendental methods and derive some
arithmetic properties of the \'etale fundamental groups of these moduli stacks.
Finally we analyse the Torelli morphism between the moduli stacks of algebraic
curves and principally polarised abelian schemes from an \'etale homotopy point
of view.
