Quantum many-body physics with large-hyperfine-spin ultracold fermions in optical lattices is at the cutting edge of ultracold atom physics and condensed matter physics. We have performed the large-scale sign-problem free quantum Monte-Carlo (QMC) simulations to study the half-filled lattice Hubbard models with SU(N) symmetry, exploring the high-symmetry states of matter in optical lattices loaded with large-hyperfine-spin ultracold fermions. We have investigated several models that are closely linked to cold atom experiments, including the SU(4) and SU(6) Hubbard models on both square and honeycomb lattices, the π-flux SU(4) Hubbard model on a square lattice, the SU(4) Hubbard model on the square lattice with a staggered pattern of flux, and the attractive SU(3) Hubbard model on a honeycomb lattice. By the QMC simulation studies of these SU(N) Hubbard models, we have found novel states of matter with large-spin fermions in optical lattices, which are not accessible or stable in solids. In addition, we have studied numerically the quantum phase transitions and finite-temperature properties of these models.