It is well-known that sparse grid algorithm has been widely accepted as an efficient tool to overcome the "curse of dimensionality" in some degree. Hyperbolic cross approximation is one of the sparse grid, whose error analysis will be discussed in this talk. The first part of this talk is devoted to the error estimate of hyperbolic cross (HC) approximations with generalized Hermite functions. The exponential convergence in both regular and optimized HC approximations has been shown. Furthermore, the error estimate of Hermite spectral method to high-dimensional linear parabolic PDEs with HC approximations has been investigated in the properly weighted Korobov spaces. In the second part of this talk, we shall give the error estimate of HC approximations with all sorts of Askey polynomials. These polynomials are useful in generalized polynomial chaos (gPC) in the field of uncertainty quantification. Under some smooth condition of the random input, the HC approximation yields exponential convergence as before. Moreover, we apply gPC to numerically solve the ordinary differential equations with slightly higher dimensional random inputs. Finally, we shall investigate the connection between the standard ANOVA approximation and Galerkin approximation, so that HC approximation can be naturally combined with ANOVA approximation. Part of this talk is the joint work with Stephen S.-T. Yau.