  We propose possible new string theories based on local world-sheet symmetries
corresponding to extensions of the Virasoro algebra by fractional spin
currents. They have critical central charges $c=6(K+8)/(K+2)$ and Minkowski
space-time dimensions $D=2+16/K$ for $K\geq2$ an integer. We present evidence
for their existence by constructing modular invariant partition functions and
the massless particle spectra. The dimension $4$ and $6$ strings have
space-time supersymmetry.
