We present new equations of state (EOS) for hydrogen and helium, intended for applications to low-mass stars (M < 1 Msun), brown dwarfs, and giant planets. They cover the range 2.10 <log T(K) < 7.06 and 4 <log P (dyn cm-2) < 19 and include new physical treatments of partial dissociation and ionization caused by both pressure and temperature effects. The hydrogen EOS is based on a careful study of nonideal interactions. In the case of helium, the principal features of the EOS physics are retained in a simplified model. The calculation is based on the free energy,minimization method, and a detailed account of the physical model has been published elsewhere. Mixtures of hydrogen and helium are obtained with the additive volume rule and an additional ideal entropy-of-mixing term. In this paper, we present extensive tabular results for both the H and He EOS, together with a critical analysis. Part of this analysis is based on a comparison with other EOS commonly used in astrophysics. Both EOS everywhere satisfy the requirements of thermal and mechanical stability and are thermodynamically consistent over most of the phase diagram. A complete set of the tables is available in the AAS CD-ROM Series, Vol. 5, and by anonymous FTP.