We measured the cross section and response functions (R_L, R_T, and R_LT) for the 16O(e,ep) reaction in quasielastic kinematics for missing energies 25 <= E_miss <= 120 MeV at various missing momenta P_miss <= 340 MeV/c. For 25 < E_miss < 50 MeV and P_miss approx 60 MeV/c, the reaction is dominated by single-nucleon knockout from the 1s1/2-state. At larger P_miss, the single-particle aspects are increasingly masked by more complicated processes. For E_miss > 60 MeV and P_miss > 200 MeV/c, the cross section is relatively constant. Calculations which include contributions from pion exchange currents, isobar currents and short-range correlations account for the shape and the transversity but only for half of the magnitude of the measured cross section.