  A subset $S$ of the Boolean hypercube $\mathbb{F}_2^n$ is a sumset if $S =
A+A = \{a + b \ | \ a, b\in A\}$ for some $A \subseteq \mathbb{F}_2^n$. We
prove that the number of sumsets in $\mathbb{F}_2^n$ is asymptotically
$(2^n-1)2^{2^{n-1}}$. Furthermore, we show that the family of sumsets in
$\mathbb{F}_2^n$ is almost identical to the family of all subsets of
$\mathbb{F}_2^n$ that contain a complete linear subspace of co-dimension $1$.
