  We compute the Brauer group of the moduli stack of elliptic curves over the
integers, localizations of the integers, finite fields of odd characteristic,
and algebraically closed fields of characteristic not $2$. The methods involved
include the use of the parameter space of Legendre curves and the moduli stack
of curves with full (naive) level $2$ structure, the study of the descent
spectral sequence in \'etale cohomology and the Leray spectral sequence in fppf
cohomology, the computation of the group cohomology of $S_3$ in a certain
integral representation, the classification of cubic Galois extensions of the
field of rational numbers, the computation of Hilbert symbols in the ramified
case for the primes $2$ and $3$, and finding $p$-adic elliptic curves with
specified properties.
