Using conformal field theory the exact critical exponents are obtained for the SU(N) generalization of the U\ensuremath{\rightarrow}\ensuremath{\infty} Hubbard model and the supersymmetric t-J model in one dimension. For the Hubbard model the critical exponent of the momentum distribution function is given as \ensuremath{\theta}=(1-1/N
)2/2 while for the supersymmetric t-J model \ensuremath{\theta} varies continuously in the range 0\ensuremath{\le}\ensuremath{\theta}\ensuremath{\le}(1-1/N
)2/2 depending on the electron concentration. The critical exponents of various correlation functions are shown to satisfy the universal scaling relations. The SU(N) t-J model in the range 0J/t\ensuremath{\le}2 belongs to the SU(N) Luttinger liquid for any finite N.
The momentum distribution and the spin-correlation functions are studied for the one-dimensional t-J model, in which the exchange interaction J is anisotropic. The discussion is focused on the nontrivial limit that J is much smaller than t. In this limit the exact ground-state wave function is a product of the spinless-fermion Slater determinant and the spin-wave function of the s=1/2 anisotropic Heisenberg antiferromagnet. With this wave function, correlation functions are calculated explicitly for finite but large systems and the nature of their singularities is studied. It is found that for the XY-like anisotropy the critical behavior of correlation functions can be described by the Tomonaga-Luttinger model, while for the Ising-like anisotropy the model seems to belong to the universality class of the Luther-Emery model of the one-dimensional electron gas.