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Several recent surprises appear dramatically to have improved the likelihoodthat the spin of rapidly rotating, newly formed neutron stars (and, possibly,of old stars spun up by accretion) is limited by a nonaxisymmetric instabilitydriven by gravitational waves. Except for the earliest part of the spin-down,the axial l=m=2 mode (an r-mode) dominates the instability, and the emittedwaves may be observable by detectors with the sensitivity of LIGO II. A reviewof these hopeful results is followed by a discussion of constraints on theinstability set by dissipative mechanisms, including viscosity, nonlinearsaturation, and energy loss to a magnetic field driven by differentialrotation.Comment: 20 pages LaTeX2e (stylefile included), 6 eps figures. Review to  appear in the proceedings of the 9th Marcel Grossman Meeting, World  Scientific, ed. V. Gurzadyan, R. Jantzen, R. Ruffin

IMPLICATIONS OF THE R-MODE INSTABILITY OF ROTATING RELATIVISTIC STARS