For four newly suggested realistic Equations of State of neutron star matter, we construct equilibrium sequences of rapidly rotating neutron stars in general relativity. The sequences are the normal and supramassive evolutionary sequences of constant rest mass. We find that for these equations of state the maximum (gravitational) mass rotating models occur (in central density and rotation rate Omega) before the maximum-Omega models. We calculate equilibrium sequences for a constant value of Omega corresponding to the most rapidly rotating pulsar PSR 1937+21. Also calculated is the radius of the marginally stable orbit and its dependence on Omega, relevant for modeling of kilo-Hertz quasi-periodic oscillations in X-ray binaries.