Density-functional calculations of the phonon spectrum and electron-phonon coupling in MgB (2) are presented. The E(2g) phonons, which involve in-plane B displacements, couple strongly to the p(x,y) electronic bands. The isotropic electron-phonon coupling constant is calculated to be about 0.8. Allowing for different order parameters in different bands, the superconducting lambda in the clean limit is calculated to be significantly larger. The E(2g) phonons are strongly anharmonic, and the nonlinear contribution to the coupling between the E(2g) modes and the p(x,y) bands is significant.