  The present paper aims to investigate the metric mean dimension theory of
continuous flows. We introduce the notion of metric mean dimension for
continuous flows to characterize the complexity of flows with infinite
topological entropy. For continuous flows, we establish variational principles
for metric mean dimension in terms of local $\epsilon$-entropy function and
Brin-Katok $\epsilon$-entropy; For a class of special flow, called uniformly
Lipschitz flow, we establish variational principles for metric mean dimension
in terms of Kolmogorov-Sinai $\epsilon$-entropy, Brin-Katok's
$\epsilon$-entropy and Katok's $\epsilon$-entropy.
