Three-dimensional homogeneous isotropic turbulence at very high Reynolds number R is studied using a variant of the Markovian eddy-damped quasi-normal theory. In the case without helicity, numerical calculations indicate the development of a $k^{-\frac{5}{3}}$ inertial range in the energy spectrum and an onset of significant energy dissipation at a time t* which appears to be independent of the viscosity v as v → 0; analytical arguments having a bearing on this behaviour, described as an ‘energy catastrophe’, are also discussed. The skewness factor (for t > t*), which increases with R, tends to 0·495 when R → ∞. When helicity is present, the existence of simultaneous energy and helicity cascades is demonstrated numerically. It is also shown that the helicity cascade inhibits the energy transfer towards large wavenumbers, in agreement with preliminary low Reynolds number results of Herring and with the conclusion of Kraichnan (1973) based on analysis of the interaction between two helicity waves. This inhibition implies a delay of the onset of energy dissipation at zero viscosity. It is shown that, whatever the relative rate of helicity and energy injection, a regime is attained at large wavenumbers k where the relative helicity tends to zero (with increasing k) and helicity is carried along locally and linearly by the energy cascade like a passive scalar. In practice, the linear regime is attained when the relative helicity is less than about 10%. The Kolmogorov constants of energy and helicity in the inertial range are determined. The impossibility of pure helicity cascades of a type conjectured by Brissaud et al. (1973a) is demonstrated. Finally it is shown that, because of dissipation and non-positive-definiteness of the helicity spectrum, non-zero total helicity may appear in the decay of unforced turbulence with zero total initial helicity, if the helicity spectrum is not initially identically zero.