We present the advantages of supercontinuum generation in chiral, therefore circularly birefringent, all-normal dispersion fibers. Due to the absence of nonlinear power transfer between the polarization eigenstates of the fiber, chiral all-normal dispersion fibers do not exhibit any polarization instabilities and thus are an ideal platform for a low-noise supercontinuum generation. By pumping a chiral all-normal dispersion fiber at 802 nm, we obtained an octave-spanning, robustly circularly polarized supercontinuum with a low noise.