We consider the stability of a circularly polarized Alfv¿n wave (the pump wave) which propagates parallel to the ambient magnetic field. Only parallel-propagating perturbations are considered, and we ignore dispersive effects due to the ion cyclotron frequency. The dissipationless MHD equations are used throughout; thus possibly important effects arising from Landau and transit time damping are omitted. We derive a series of analytical approximations to the dispersion relation using A=(ΔB/B0)2 as a small expansion parameter; ΔB is the pump amplitude, and B0 is the ambient magnetic field strength. We find that the plasma &bgr; (the square of the ratio of the sound speed to the Alfv¿n speed) plays a crucial role in determining the behavior of the parametric instabilities of the pump. If 01, we find &ggr;max∝A3/2, while if &bgr;≈0, we obtain &ggr;max∝A1/3; moreover, if &bgr;≈0 there is a nearly purely growing instability. In contrast to the familiar decay instability, for which the backward propagating Alfv¿n wave has lower frequency and wavenumber than the pump, we find that if &bgr;>1 the instability is really a beat instability which is dominated by a transverse wave which is forward propagating and has frequency and wavenumber which are nearly twice the pump values. Only the decay instability for 0<&bgr;<1 can be regarded as producing two recognizable normal modes, namely, a sound wave and an Alfv¿n wave. We discuss how the different characteristics of the instabilities may affect the evolution of Alfv¿n waves in the solar wind. However, for a solar wind in which &bgr;≈1 the growth times of the instabilities are probably too long for these instabilities to have an appreciable effect inside 1 AU. ¿ American Geophysical Union 1993