The magnetohydrodynamic evolution of the interaction region between the inner edge of an accretion disk and the magnetosphere of the central object is studied by means of time-dependent numerical simulations. The simulations assume the disk is adiabatic, is axisymmetric, has nonzero resistivity, and is initially in Keplerian rotation. The magnetosphere is assumed to be initially in magnetostatic equilibrium, corotating with the central star, and threaded by one of three different initial magnetic field topologies: (1) a pure dipole field, which also threads the disk continuously everywhere; (2) a dipole field excluded from the disk by surface currents; and (3) a dipole field continuously threading a disk superposed with a uniform axial magnetic field. A number of exploratory simulations are performed by varying the field strength, the disk density and inner radius, the magnitude of the resistivity, and the stellar rotation rate. These simulations are designed as an initial study of the magnetohydrodynamics of the interaction region. Generally, we find that rapid evolution of the disk occurs because of angular momentum transport by either the Balbus-Hawley instability or magnetic braking effects. Equatorial accretion results on a dynamical timescale unless the magnetic pressure of the magnetosphere exceeds the ram pressure of the accreting disk plasma; the latter we find to be a highly time-dependent quantity. In the case of a pure dipole magnetospheric field, however, rapid stellar rotation can result in a field geometry that inhibits polar accretion even when ram and magnetic pressures balance. In contrast, we find that polar accretion can occur regardless of the stellar rotation rate when strong global disk magnetic fields combine with stellar magnetic fields to create a favorable net field topology. Highly time-dependent winds are evident in the evolution of all three field topologies. The winds are generally channeled along field lines that have been opened through reconnection. The speed and variability of the outflows is dependent on the magnetic field strength and accretion topology. Net torque on the star during accretion is measured to be positive, i.e., the star is being spun up.