  We develop mean dimension theory for $\mathbb{R}$-flows. We obtain
fundamental properties and examples and prove an embedding theorem: Any real
flow $(X,\mathbb{R})$ of mean dimension strictly less than $r$ admits an
extension $(Y,\mathbb{R})$ whose mean dimension is equal to that of
$(X,\mathbb{R})$ and such that $(Y,\mathbb{R})$ can be embedded in the
$\mathbb{R}$-shift on the compact function space $\{f\in
C(\mathbb{R},[-1,1])|\;\mathrm{supp}(\hat{f})\subset [-r,r]\}$, where $\hat{f}$
is the Fourier transform of $f$ considered as a tempered distribution. These
canonical embedding spaces appeared previously as a tool in embedding results
for $\mathbb{Z}$-actions.
