  Local Fourier trnasforms, analogous to the $\ell$-adic Fourier transforms,
are constructed for connections over $k((t))$. Following a program of Katz, a
meromorphic connection on a curve is shown to be rigid, i.e. determined by
local data at the singularities, if and only if a certain cohomological
condition is satisfied. We show that the index of rigidity is invariant under
Fourier transform.
