  Here a new notion of fractional length of a smooth curve, which depends on a
parameter $\sigma$, is introduced that is analogous to the fractional perimeter
functional of sets that has been studied in recent years. It is shown that in
an appropriate limit the fractional length converges to the traditional notion
of length up to a multiplicative constant. Since a curve that connects two
points of minimal length must have zero curvature, the Euler--Lagrange equation
associated with the fractional length is used to motivate a nonlocal notion of
curvature for a curve. This is analogous to how the fractional perimeter has
been used to define a nonlocal mean curvature.
