The spectral weight A (p, ω) of the two-dimensional t-J and Hubbard models has been calculated using exact diagonalization and quantum Monte Carlo techniques, at several densities 0.5≤< n>≤ 1.0. The photoemission (ω< 0) region contains two dominant distinct features, namely a low-energy quasiparticle peak with bandwidth of order J, and a broad valence band peak at energies of order t. This behavior persists away from half-filling, as long as the antiferromagnetic correlations are robust. The results give support to theories of the copper oxide materials based on the behavior of holes in antiferromagnets, and they also provide theoretical guidance for the interpretation of experimental photoemission data for the cuprates.