By using measurements with the University of Iowa plasma wave experiment on the Imp 6 satellite a study has been conducted of the spectrum of electrostatic plasma waves in the terrestrial magnetosheath. Electrostatic plasma wave turbulence is almost continuously present throughout the magnetosheath with broadband (20 Hz to 70 kHz) rms field intensities typically 0.01--1.0 mV m-1. Peak intensities of about 1.0 mV m-1 near the electron plasma frequency (30--60 kHz) have been detected occasionally. Two or three components can usually be identified in the spectrum of magnetosheath electrostatic turbulence: a high-frequency (?30kHz) component peaking at the electron plasma frequency fpe, a low-frequency component with a broad intensity maximum below the nominal ion plasma frequency fpi (~ fpe/43), and a less well defined intermediate component in the range fpi <f<fpe. The intensity distribution of magnetosheath electrostatic turbulence clearly shows that the low-frequency component is associated with the bow shock, suggesting that the ion heating begun at the shock continues into the downstream magnetosheath. Electrostatic waves below 1 kHz are polarized along the magnetic field direction, a result consistent with the polarization of electrostatic waves at the shock. The high- and intermediate-frequency components are features of the magnetosheath spectrum which are not characteristic of the shock spectrum but are often detected in the upstream solar wind. The intensity distribution of electrostatic turbulence at the magnetosheath plasma frequency has no apparent correlation with the shock, indicating that electron plasma oscillations are a general feature of the magnetosheath. The plasma wave noise shows a tendency to decrease toward the dawn and dusk regions, consistent with a general decrease in turbulence away from the subsolar magnetosheath. Occasionally, intense electrostatic bursts occur which are correlated with magnetic field gradients and fluctuations, possibly indicating the presence of strong current systems such as can occur at the magnetopause.