The current convective instability is identified as the dominant mechanism for producing turbulence and transport in the magnetopause current layer. A linear stability analysis demonstrates that the instability should be robustly unstable for the parameters of the magnetopuase current layer. Three-dimensional nonlinear simulations indicate that the current layer evolves to a strongly turbulent state. The current convective instability is driven unstable by the cross-field gradient of the parallel current in the collisionless plasma. A simple physical picture of the instability is presented. For narrow current layers with scale length L less than an ion Larmor radius &rgr;i, the growth rates exceed the ion gyrofrequency even with Ti≫Te. A set of nonlinear fluid equations are derived in the unmagnetized ion limit &rgr;i>L. In three-dimensional simulations based on these equations the current layer evolves to a turbulent state with layers of extended but irregular clifflike structures with transverse scale length of the order of &rgr;es, the electron Larmor radius based on Ti. The anomalous cross-field transport rate D⊥ of current resulting from this instability is given by D⊥~&rgr;esvy with vy the electron parallel streaming velocity. The present linear theory and simulations are limited to electrostatic disturbances. The implications of these results for understanding the structure of the magnetopause current layer are discussed. ¿ American Geophysical Union 1994