This thesis presents research into parallel linear solvers for block-diagonal-bordered sparse matrices, but first we review algorithms for sequential sparse linear solvers. We address both direct and iterative solvers in this work, each having their own advantages and disadvantages. Direct methods obtain the exact solution for a series of simultaneous linear equations in a finite number of operations, whereas iterative methods calculate sequences of approximations that may or may not converge to the solution. While direct methods obtain an exact solution of the linear system, they may require significantly more computations than required for iterative methods to obtain a usable solution. The remainder of this chapter discusses two related direct methods, LU factorization and Choleski factorization, and one iterative solver, the Gauss-Seidel method.