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We present three-dimensional magnetohydrodynamical simulations of the nonlinear evolution of the magnetorotational instability (Balbus & Hawley 1991) in an initially isothermal, vertically stratified accretion disk. The simulations are local in the plane of the disk, but are global in the vertical direction in the sense that the computational domain encompasses several scale heights. We find that the instability generates and maintains magnetohydrodynamic turbulence in stratified disks. The properties of the saturated, turbulent state are similar to those reported for nonstratified homogeneous boxes, e.g., the power-law spectral distribution is consistent with Kolmogorov. The angular momentum transport rate is proportional to the magnetic pressure; if the average rate is scaled to the equatorial pressure, as in a standard "α-disk" model, the constant of proportionality is α ≲ 0.01 for most of our simulations. This value may be limited somewhat by the numerical resolution of our simulations. We find buoyancy does not play an important role as a saturation mechanism. The nonlinear evolution of disks that begin with a variety of initial magnetic field geometries and strengths is similar. In particular, models in which the most unstable wavelengths are initially well resolved saturate at and maintain roughly the same magnetic energy density, suggesting dynamo action in the disk. <P />We use the simulations to study the effect of the instability on the vertical structure of magnetic accretion disks. After saturation, the disk consists of a weakly magnetized core surrounded by a strongly magnetized corona. Changing the vertical boundary conditions does not significantly alter this structure. The vertical flux of magnetic energy is small compared to the local magnetic energy generation and dissipation rates. The disk is stable to Parker and buoyancy modes. Models evolved with an adiabatic equation of state undergo substantial heating due to nonlinear dissipation, resulting in an increase in the scale height of the disk. The vertical structure produced by the instability, particularly the presence of a strongly magnetized corona, may have relevance to the production of MHD winds from disks.