Observations from the Earth's geomagnetic tail have established that the current sheet is often bifurcated with two peaks in the current density. These so-called bifurcated current sheets have also been reported in a variety of simulations and often occur in conjunction with significant temperature anisotropy. In this work, a new self-consistent Vlasov equilibrium is developed that permits both the current profile and temperature anisotropy to be independently adjusted. This new equilibrium has a sufficient flexibility to model a wide variety of bifurcated sheets observed in both kinetic simulations and space observations, and transforms continuously back to the standard Harris sheet model with a single peak in the current density. The linear stability of these layers with respect to the tearing mode is examined in the framework of resistive MHD and full Vlasov theory. From the simplified fluid theory, it is demonstrated that a bifurcated current profile has a stabilizing influence on the resistive tearing instability. However, the resistive MHD model is not really appropriate to model the highly collisionless plasma conditions in the magnetosphere. To obtain reliable predictions, Vlasov theory is required and the approach in this thesis employs both standard analytic techniques and a formally exact treatment in which the full orbit integral is numerically evaluated. The resulting linear growth rate for the collisionless tearing instability and the mode structure are verified with 2D full kinetic particle-in-cell simulations. The simplified analytic theory is reasonably accurate in capturing these dependencies for long wavelength modes, but the short wavelength regime generally requires the full numerical treatment to accurately compute the growth rate. The results from these different approaches consistently demonstrate that a bifurcated current profile has a strong stabilizing influence on the collisionless tearing mode in comparison to centrally peaked layers with a similar thickness. In collisionless tearing, electron temperature anisotropy is strongly destabilizing in the limit T_e⊥ > T _e∥ and strongly stabilizing when T _e⊥ < T_{e ∥}. Thus, the collisionless tearing instability is determined by the competition between these two influences.