ISEE 1 and 2 magnetic field measurements are used to examine the structure of the low beta, quasi-perpendicular shock. A shock crossing database consisting of ISEE 1 satellite crossings from the beginning of the mission in 1977 to the end of 1980 is utilized to identify shock crossings for this study. A set of 20 low beta, quasi-perpendicular shock crossings are drawn from the database for study. Analysis of the shock overshoots indicates that the strength of the overshoot of low beta, quasi-perpendicular shocks increases as the ratio of the Mach number to the first critical Mach number (or ratio of criticality) increases. There are subcritical crossings which have nonnegligible overshoots and other subcritical crossings which exhibit no overshoot. Wave analysis shows that the power of the downstream waves also increases as a function of this ratio of criticality. Upstream of the shock, large-amplitude, low-frequency whistler mode and higher-frequency (f approximately 1 Hz) whistler waves are evident for subcritical and marginally critical shocks. The lower-frequency whistlers are right-hand elliptically polarized and phase stand upstream of the shock, propagating along the shock normal direction. The thickness of the shock is found to be within a factor of 1 and 2 times greater than the wavelength of this precursor wave. This result is inconsistent with the conjecture that the shock is merely the last amplified cycle of the precursor wave, for if this were true, the thickness of the shock from minimum to maximum would be one half of the precursor wavelength. The 1-Hz waves are right-hand elliptically polarized and propagate upstream obliquely to the magnetic field direction. Downstream of the marginally critical and supercritical shock, left-hand elliptically polarized waves are found to propagate along the magnetic field direction and have frequencies of about 0.2-0.8 f(ci). These ion cyclotron waves appear to result from the excitation of the Alfven ion cyclotron (AIC) instability. The AIC instability is driven by the T(i perpendicular-to) > T(i parallel-to) temperature anisotropy created in front of the shock by the reflection of solar wind ions. Ion cyclotron waves act to pitch angle scatter the ions downstream of the shock and remove the temperature anisotropy. A transitional behavior in the noncoplanar component of the magnetic field occurs at or about the first critical Mach number. Below the critical Mach number, the noncoplanar component is associated with the upstream whistler train. When the ratio of criticality is approximately unity, the noncoplanar component is isolated from any upstream or downstream wave activity. In the supercritical regime, this component of the field is associated with the downstream ion cyclotron wave train. For all ranges of criticality, the noncoplanar component is seen to lie within the shock ramp, and the transitional behavior of this component of the field indicates that it is an inherent part of the shock.