Weighted essentially non-oscillatory (WENO) schemes are a popular class of high order accurate numerical methods for solving hyperbolic conservation laws. They have been applied extensively in computational fluid dynamics and other scientific problems. However, for complicated multidimensional problems, it often leads to large amount of operations and computational costs in the numerical simulations by using nonlinear high order accuracy WENO schemes such as fifth order WENO schemes. How to achieve fast simulations by high order WENO methods for solving hyperbolic conservation laws is a challenging and important question. In this talk, I shall present our recent work on applying fast sweeping methods and sparse-grid techniques for efficient computations of WENO schemes. Fast sweeping methods are a class of efficient iterative methods for solving steady state problems of hyperbolic PDEs. They utilize alternating sweeping strategy to cover a family of characteristics in a certain direction simultaneously in each sweeping order. Coupled with the Gauss-Seidel iterations, these methods can achieve a fast convergence speed for computations of steady state solutions of hyperbolic PDEs. We design absolutely convergent fixed-point fast sweeping WENO methods for solving steady state solutions of hyperbolic conservation laws. For high-dimensional problems, sparse-grid techniques are efficient approximation tools to reduce degrees of freedom in the discretizations. We apply the sparse-grid combination technique to fifth order WENO finite difference schemes for solving time dependent hyperbolic PDEs defined on high spatial dimension domains. Extensive numerical experiments shall be shown to demonstrate large savings of computational costs by comparing with simulations using traditional methods for solving hyperbolic conservation laws.