Twenty-two states in U-238 have been observed with 18 MeV He-4 ions on a thick target. Eight 2 + states between 966 and 1782 keV and three 3- states are populated by direct E2 and E3, respectively. The remaining states are either weakly excited by multiple Coulomb excitation and/or populated by the y-ray decay of the directly excited states. Spin assignments are based on gamma-ray angular distributions. Reduced transition probabilities have been deduced from the gamma-ray yields. The B(E2) values for excitation of the 2+ states range from 0.10 to 3.0 W.u. (281 W.u. for the first 2+ state). For the 3 states, the B(E3, 0 --> 3-) values are 7.1, 7.8, and 24.2 W.u. Several of the 2+ states have decay branches to the one-phonon states with B(E2) values between 27 and 56 Wu. which are an order of magnitude larger than the B(E2) values between the one- and zero-phonon states. This disagrees with our present understanding of collectivity in nuclei if these 2+ states are considered to be collective two-phonon excitations. However, the excitation energies of these 2+ states with respect to the one-phonon states are only 1.3 to 1.6. The B(E1) values for 17 transitions between the positive- and negative-parity states range between 10(-3) and 10(-7) W.u. The B(E1) branching ratios for many of these transitions have large deviations from the Alga-rule predictions. These deviations can be understood by the strong Coriolis coupling between the states of the one-phonon octupole quadruplet in deformed nuclei. The general features of the experimental results for the B(E3) values are reproduced by the microscopic calculations of Neergard and Vogel when the Coriolis coupling between the states of the octupole quadruplet is included.