In this thesis, we study circulant-type preconditioners for solving Toeplitz systems by using Krylov subspace methods. A Krylov subspace method for solving the system Ax = b, is an iterative method which seek an approximation solution xₘ of x from the m-dimensional affine subspace x₀ + Kₘ(A, r₀)where x₀ is an initial guess to the solution, r₀= b- Ax₀ is the initial residual and Kₘ(A, r₀) ≡ Span{r₀,Ar₀,A²r₀, ... , Aᵐ⁻¹r₀). The conjugate gradient (CG) method and the generalized minimal residual (GMRES) method are two well known methods among Krylov subspace methods and are considered in this thesis. Some basic theory of these methods and circulant-type preconditioners are given in Chapter 1. In Chapter 2, we consider ill-conditioned Toeplitz systems. Usually, the convergence rate of the CG method is slow for this kind of systems. To deal with this problem, Potts and Steidl proposed in [30] an {ω]-circulant preconditioner for accelerating the convergence rate. It was shown that the CG method applied in solving the preconditioned system converges superlinearly and the total operation cost is O(n log n) where n is the size of the system. We generalize the preconditioner in [30] for block Toeplitz systems. We show the clustering property of the preconditioned matrix and give the total opertation cost of the method. Numerical examples and an application in image restoration problems are given. In Chapter 3, we consider the initial value problem (IVP) of linear constant coefficient differential-algebraic equations (DAEs), Ax'(t) + Bx(t) = f(t) where A, B are square matrices and A is singular. If det(λA + B) with λ ∈ C is not identically zero, the system of DAEs is solvable and can be separated into two uncoupled subsystems. One of them can be solved analytically and the other one is a system of ordinary differential equations (ODEs). We discretize the ODEs by boundary value methods (BVMs) and solve the linear system by using the GMRES method with Strang-type block-circulant preconditioner. It was shown that the preconditioner is nonsingular when the BVM is Aₖ₁,ₖ₂-stable, and the sequence of preconditioned matrices has clustered spectra. Therefore, the number of iterations for solving the preconditioned systems by the GMRES method is bounded by a constant that is independent of the discretization mesh. Numerical results are also given.