We derive general expressions for the LMC velocity field which we fit to kinematical data for 1041 carbon stars. We demonstrate that all previous studies of LMC kinematics have made unnecessary over-simplifications that have led to incorrect estimates of important structural parameters. We compile and improve LMC proper motion estimates to support our analysis. We find that the kinematically determined position angle of the line of nodes is 129.9 +/- 6.0 deg. The LMC inclination changes at a rate di/dt = -103 +/- 61 deg/Gyr, a result of precession and nutation induced by Milky Way tidal torques. The LMC rotation curve V(R) has amplitude 49.8 +/- 15.9 km/s, 40% lower than what has previously (and incorrectly) been inferred from e.g. HI. The dynamical center of the carbon stars is consistent with the center of the bar and the center of the outer isophotes, but not with the HI kinematical center. The enclosed mass inside 8.9 kpc is (8.7 +/- 4.3) x 10^9 M_sun, more than half of which is due to a dark halo. The LMC has a larger vertical thickness than has traditionally been believed. Its V/sigma is less than the value for the Milky Way thick disk. We discuss the implications for the LMC self-lensing optical depth. We determine the LMC velocity and orbit in the Galactocentric rest frame and find it to be consistent with the range of velocities that has been predicted by models for the Magellanic Stream. The Milky Way dark halo must have mass >4.3 x 10^{11} M_sun and extent >39 kpc for the LMC to be bound. We predict the LMC proper motion velocity field, and discuss techniques for kinematical distance estimation. [ABRIDGED]

57 pages, LaTeX, with 11 PostScript figures. Submitted to the Astronomical Journal